logikis minimaluri kursi - bpa.gebpa.ge/book/book05.pdf · Sesavali: ras swavlobs logika da ris...

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daviT TevzaZe

logikis minimaluri kursi

damxmare saxelmZRvanelo IB –evrokavkasiuri universitetis studentebisaTvis

Tbilisi 2009

2 Sinaarsi

winasityvaoba

Sesavali: ras swavlobs logika da ris codnas gvaZlevs es wigni?

Tavi I. zogadi cnobebi simravleTa Teoriis Sesaxeb

$1. simravleebi da maTi elementebi. qvesimravle. carieli

universaluri da erTeulovani simravleebi. venis diagramebi.

$2. ZiriTadi damokidebulebebi simravleebs Soris.

$3. operaciebi simravleebze

$4. usasrulo simravleebi da maTTan dakavSirebuli zogierTi

siZnele. raselis paradoqsi.

Tavi II. propoziciuli logikis sawyisi cnebebi da ZiriTadi kanonebi

$1. ras vuwodebT propoziciul logikas? martivi da rTuli

winadadebebi. sawinadadebo makavSireblebi.

$2. winadadebaTa WeSmaritebiTi mniSvneloba. WeSmaritebiTi cxrilebi.

$3. damokidebulebani sawinadadebo makavSireblebs Soris.

$8. propoziciuli logikis ZiriTadi kanonebi

Tavi III. predikatTa logikis sawyisi cnebebi da ZiriTadi kanonebi

$1. ras vuwodebT predikatTa logikas?

$2. sawinadadebo formebi, cvladebi da konstantebi.

$3. kvantorebi da kvantorebian winadadebaTa Tvisebebi.

damateba 1. cota ram dasabuTebis Sesaxeb

damateba 2. savarjoSoebi

damateba 3. ra wavikiTxoT?

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winaswasityvaoba

wignis mizania gauadvilos studentebs zogierTi im Temis gaazreba,

romelic, rogorc praqtikam gvaCvena, garkveul wilad, iwvevs sirTuleebs

logikis standartuli kursis aTvisebisas. is SeiZleba gamoyenebul iqnas

rogorc damoukideblad, aseve l.mWedliSvilisa da n.ivaniZis

saxelmZRvanelosTan erTad (ix.: l.mWedliSvili, n.ivaniZe. logika. Tbilisi,

2003w.). saTaurSi sityva “minimaluri” niSnavs, rom aq gadmocemuli masala

warmoadgens codnis im aucilebel minimums, romelic umciresi dadebiTi

SefasebisTvisaa saWiro.

4 Sesavali: ras swavlobs logika da ris codnas gvaZlevs es

wigni?

logika, iseve rogorc sxva mecnierebebi, daadgens WeSmaritebebs. amitom,

rodesac vsvamT kiTxvas, ras swavlobs logika, pirvel rigSi vgulisxmobT

imas, Tu ra saxis WeSmaritebebzea saubari logikaSi, gansxvavebiT, vTqvaT,

fizikis, maTematikis, anda Tundac istoriis WeSmaritebebisgan.

naTqvamSi rom gaverkveT, yuradReba mivaqcioT Semdegs:

(1). fizika, zogadad, aRwers bunebas, da miCneulia, rom am aRweris Sedegad

daadgens im kanonzomierebebs, romelic fizikur sinamdvileSi moqmedebs.

amgvarad, is debulebebi, romelic fizikis ZiriTad Sinaars Seadgens,

erTgvarad, asaxavs imas, rac fizikur samyaroSi xdeba. rodesac

eqsperimentiT, anda sxva gziT vrwmundebiT, rom romelime debuleba zustad

asaxavs am xdomilebebs, Cven saqme gvaqvs WeSmaritebasTan, romelsac

fizikis mecniereba daadgens.

(2). analogiurad, maTematikac aRwers raRac sinamdviles, romelic

bunebisgan gansxvavebiT, aramaterialuria im azriT, rom mis obieqtebs, ase

vTqvaT, xeliT ver “SevexebiT” imdagvarad, rogorc es fizikuri obieqtebis

SemTxvevaSi SegviZlia gavakeToT. amitomac, rTulia movZebnoT samyaroSi

maTematikis debulebaTa Sesatyvisi iseTi xdomilebebi, romelic maTi

adeqvaturobis sazomi gaxdeboda. Tumca, maTematikuri Teoriebis agebisas,

Cven garkveuli principebiT vxelmZRvanelobT da Tu ki Cven mier

gamoTqmuli debuleba TanxmobaSiaMam principebTan, vityviT, rom

dagvidgenia maTematikuri WeSmariteba. unda aRiniSnos, rom xSirad es

Tanxmoba sakmaod rTuli dasanaxia da Cveni gonebrivi resursis srul

mobilizebas gulisxmobs. sabolood, aRniSnuli Tanxmoba maTematikuri

dafuZnebis saxes iRebs da Tu ki es dafuZneba unakloa dasabuTebis

TvalsazrisiT, SegviZlia Tamamad vTqvaT, rom Cveni debuleba WeSmaritia

Tavdapirvelad dadgenili principebiT marTul sivrceSi.

5 (3). sxvagvaradaa saqme istoriuli WeSmaritebebis dadgenisas. aq Cven mier

gamoTqmuli winadadebebi TanxmobaSi unda iyos imasTan, rac sazogadoebis

ganviTarebis garkveul etapze, ama Tu im xalxis garSemo xdeboda.

sirTules is garemoeba qmnis, rom xSirad, Cven sxvaTa damowmebiT, anu im

pirTa aRwerebis saSualebiT vmsjelobT, romlebic am movlenaTa

Tanamedroveni iyvnen, an am movlenaTa Sesaxeb sxvaTa monaTxrobiT ician.

sxva sityvebiT, Cven saSualeba ara gvaqvs Cvens uSualo gancdas

daveyrdnoT, anda daveyrdnoT eqsperiments, rogorc fizikaSi, an

dasabuTebas, rogorc maTematikaSi. es ki sxvadasva interpretaciis

saSualebas iZleva, rac, xSirad, istorikosebs Soris davis safuZvels

warmoadgens.

Tumca istoriul mecnierebasac gaaCnia garkveuli xerxebi, rom codna,

romelsac is ganapirobebs, sandoobis ufro maRali xarisxi hqondes, vidre

es ubralo varauds gaaCnia. aseTia, rogorc arqeologiis monacemTa

analizi, aseve im teqstebis sandoobis Semowmebis mTodologiebi, romelic,

rogorc istoriuli wyaroebi, SemogvrCa, da romelTa damuSavebis meTodika

droTa manZilze ixveweba. saistorio wyaroebad SegviZlia miviCnioT ara

mxolod Txroba romelime ambisa, aramed im periodis dokumentebi, romelic

mocemuli drois monakveTSi adamianTa Soris urTierTobebis

daregulirebisTvis iyo Seqmnili: brZanebebi, kanonTa krebulebi,

sasamarTlo oqmebi Tu sxva Canawerebi, xelSekrulebebi, angariSebi,

saSemosavlo dokumentebi da bevri sxva, rasac dRes Cven, SesaZloa,

yuradRebasac arc vaqcevT.

Tu amgvar msjelobas gavyvebiT, vnaxavT, rom yoveli mecniereba, garkveul

wilad, Cveni samyaros Sesaxeb codnas ganapirobebs imiT, rom yvelafers,

rac Cvens garSemoa, WeSmaritad miCneuli winadadebebiT aRwers. Tumca,

WeSmaritad miCnevis safuZvlebis sxvadasxvaobis gamo, is sxvadsvagvaria

Tavisi sandoobis xarisxis mixedviT. im meTodebis wyalobiT, rasac fizika

eyrdnoba, fizikur movlenaTa codna gacilebiT ukeT aris dafuZnebuli,

6 vidre, igive istoriuli mecnierebisa. magram kiTxva Cndeba TviTon codnisa

da kargad dafuZnebuli codnis Sesaxeb. ra aris es? rogor aris

SesaZlebeli WeSmarit winadadebaTa WeSmaritebis xarisxis gansazRvra da

ra aris Cveni rwmenis safuZveli, rodesac raimeze vambobT, rom is

WeSmaritia? metic, ra pirobebs unda akmayofilebdes Cveni codna, rom masze

vTqvaT es mecnieruli codnaa da ganvasxvavoT is banalur

WeSmaritebaTagan, romelic Cvens yoveldRiurobas aRwers? magaliTad,

WeSmaritebaa “e=mc2

” da Tu ki dRes mze anaTebs, WeSmaritia agreTve “dRes

kargi amindia”. ra qmnis am codnaTa Soris sxvaobas da ra aris is, rac

mecnierul codnas dasabuTebulobis met xarisxs aniWebs? sxva sityvebiT,

ra aris dasabuTeba da rogoria misi buneba? es gaxlavT ZiriTadi kiTxva,

romelsac logika, rogorc mecniereba, pasuxobs. anu, logika ikvlvevs da

eZiebs imas, Tu ra iqneboda WeSmariti dasabuTebis Sesaxeb, Tu ki

dasabuTebas Seswavlis sagnad Tavad vaqcevdiT.

rac Seexeba logikas, rogorc saswavlo disciplinas, misi amocanaa, ukeTes

SemTxvevaSi, gagvacnos Tu ras miaRwia am mimarTulebiT kacobriobam da ra

progress SeiZleba velodeT kidev. xolo logikis minimaluri kursis

daniSnulebaa daveufloT logikur teqstebTan muSaobis Cvevas, pirvel

rigSi, igulisxmeba saxelmZRvaneloebi.

7 Tavi I. zogadi cnobebi simravleTa Teoriis Sesaxeb

$1. simravleebi da maTi elementebi. qvesimravle. carieli, universaluri da

erTeulovani simravleebi. venis diagramebi.

sityvas simravle Cven vixmarT saganTa nebismieri erTobliobis aRsawerad,

romlebic raRac niSniT arian gaerTianebuli. magaliTad, Cven SegviZlia

vilaparakoT klasSi myof studentTa simravleze, anda amave klasSi myof

gogonaTa simravleze. Tu nagulisxmev saklaso oTaxs ar gavcdebiT,

SegviZlia aseve vilaparakoT skamebis simravleze, romelic am saklaso

oTaxSia, anda simravleze romlic mxolod saklaso oTaxSi myofi

sagnebisagan aris Sedgenili. maSin, masSi moeqceva rogorc skamebi, aseve

magidebi (an merxebi), dafa, wignebi, studentTa xelCanTebi, carci an

markeri da a.S., anu yvelaferi is, rasac SegviZlia xeli movkidoT da

amJamad aq, am saklaso oTaxSi imyofeba. saganTa sxvadasxvaobis miuxedavad,

maT saerTo ram Tviseba gaaCniaT: isini amJamad da aq, am saklaso oTaxSi

arian Tavmoyrili. metic, imave saklaso oTaxis SigniT Cven kidev gvaqvs

saSualeba simravlis gafarToebisa, Tu ki masSi myof sulier da usulo

saganTa erTobliobas ganvixilavT. es erToblioba, sul cota erT

damatebiT obieqts Seicavs, romelic aqamde ar gvixsenebia, kerZod,

profesors, romelic amJamad studentebs, SesaZloa, leqcias utarebdes.

saganTa simravleebi, romlebic am saklaso oTaxiT SemovfargleT,

gansxvavebulnia im azriT, rom maTi Semadgeneli obieqtebi, anu is sagnebi,

romlebic erTobliobebs qmnian, Tavad arian gansxvavebulni. magaliTad, am

klasSi myofi skamebis simravle gansxvadeba rogorc am klasSi myofi

magidebis simravlisagan, aseve am klasSi myofi studentebis simravlisgan.

sainteresoa, rom es gansxvaveba, SesaZloa, raodenobrivic aRmoCndes,

vinaidan dasaSvebia, rom ganxilvis momentSi klasSi saerTod ar

imyofebodes arc erTi studenti, anda imyofebodes imaze meti, vidre skami.

Tu Cvens msjelobas odnav ganvavrcobT, vnaxavT, rom simravleebi Tavad

SeiZleba iqcnen im obieqtebad, romelTa simravled ganxilva iseve azriani

iyos, rogorc skamebis simravlisa. magaliTad, imave saklaso oTaxis

farglebs rom ar gavcdeT, SegviZlia vTqvaT, rom am klasSi arsebuli

yvela sagnis simravle emTxveva simravles, romelsac erToblovad qmnian

masSi myofi skamebis, magidebis, dafebis, qal-vaJi studentebis, maTi piradi

nivTebis, dafebis, markerebis, carcis naWrebis da a.S. simravleebi. cxadia,

saganTa simravleebis daxasiaTebisas am saganTa Tvisebebs, ra Tqma unda

mniSvneloba aqvs, magram, meores mxriv, zogjer, yvela maTi sxva Tviseba,

garda amJamad am simravlis sagnad yofnisa, garkveul wilad, mniSvnelobas

kargavs. magaliTad, me Tu minda am klasSi myofi studentebis simravlis

Sesaxeb (rac igivea, am klasSi myof studentebze) raime zogadi msjeloba

gamovTqva, CemTvis, cxadia, sul erTi unda iyos qalia es studenti, Tu vaJi.

8 amiT Cven imis Tqma gvinda, rom xSirad aranairi mniSvneloba ara aqvs imas,

Tu ra sagnebisgan Sedgenil simravleebs vixilavT, aramed mniSvnelovania

is, rom es sagnebi garkveul erTobliobas qmnian da maTze msjeloba

erToblivadaa SesaZlebeli, rogorc simravleze. saklaso oTaxis

farglebs rom gavcdeT, simravleTa magaliTebis motana mravlad SeiZleba:

kavkasionis mwvervlTa simravle, aziis kontinentis tbebis simravle,

evropuli qalaqebis quCebis simravle, amerikaSi mcxovrebi polonelebis

simravle da a.S. gansakuTrebiT mosaxerxebelia saubari ricxvTa

simravleeze da amis uamravi magaliTia CvenTvis cnobili standartuli

maTematikis kursidan: naturalur ricxvTa simravle, namdvil ricxvTa

simravle, luw ricxvTa simravle, im ricxvTa simravle romelic 100-ze

naklebia da unaSTod iyofa 3-ze da a.S. amgvarad, ra sagnebi Seadgens

simravles, erTgvarad, naklebmniSvnelovani xdeba, Tu ki Tavad

simravleebze vapirebT saubars. ra Taviseburebani gaaCnia simravleebs, ra

damokidebulebani SeiZleba arsebobdes maT Soris, rogor SeiZleba

simravleebiT operireba, anda ra gamoyeneba SeiZleba moeZebnos maTi

Tvisebebis aRweras, is kiTxvebia, razedac pasuxs mecnierebis (ufro

zustad, maTemetikuri mecnierebis) erT-erTi dargi – simravleTa Teoria

iZleva. imisaTvis, rom, garkveulwilad, gagviadvildes simravleebze

msjeloba, an simravleebiT operireba, SevTanxmdeT Semdegze:

1.1. sagans, romelic raime simravlis obieqtadaa moazrebuli, am

simravlis elementi ewodeba. elementi, nebismier simravlesTan

mimarTebaSi xasiaTdeba imiT, rom is am simravles an ekuTvnis, an ar

ekuTvnis. magaliTad, me Tu studenti var, romelic am saklaso

oTaxSi imyofeba, me var am klasSi myofi studentebis simravlis

elementi da amgvarad, vekuTvni am simravles. meores mxriv, rogorc

aseTi, me ar vekuTvni am klasSi myofi skamebis simravles. sazogadod,

miRebulia, rom simravleTa aRsaniSnavad ixmaron laTinuri anbanis

didi asoebi, indeqsebiT an maT gareSe (mag., A, B, C, . . . A1, B1, C1, . . .). simravlis elementebis zogadi aRniSvnisaTvis ixmareba amave anbanis

patara asoebi, aseve indeqsebiT, an maT gareSe am anbanis bolodan

(mag ., x, y, z, . . .x₁, y₁ , z₁ . . .), xolo anbanis sawyisi patara asoebi

(indeqsebiT, an maT gareSe), gamoiyenebulia mocemul simravleSi konkretuli sagnis misaTiTeblad. imisaTvis rom mikuTvneba

gamovxatoT, ixmareba niSani ∈ xolo rodesac amas ara aqvs adgili,

ixmareba niSani ∉. sxva sityvebiT, Canaweri x∈A ikiTxeba rogorc “x aris

A simravlis elementi”, xolo Canaweri x∉A - “x ar aris A simravlis

elementi”.

1.2. iTvleba, rom simravle mocemulia, Tu ki mocemulia am simravlis

yvela elementis CamonaTvali, anda gansazRvrulia wesi, romliTac

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nebismieri mocemuli elementisaTvis calsaxad SegviZlia

ganvsazRvroT aris Tu ara is am simravlis elementi. meore

gansakuTrebiT mniSvnelovania, rodesac saqme iseT simravleebTan

gvaqvs, romlebic Zalian bevr elements Seicavs, anda maTi raodenoba

usasruloa. magaliTad, mzis sitemaSi myofi asteroidTa simravle,

savaraudod, sasrulia, Tumca misi CamonaTvalis saxiT mocema, albaT,

SeuZlebeli iqneboda. meores mxriv, mzis sistemis planetaTa

simravlis gansazRvra SesaZlebelia arsebuli (an ukve cnobili)

planetebis saxelebis CamoTvliT. sxvaTa Soris, mzis sistemis

planetaTa simravle da CvenTvis cnobili planetebis simravle,

SesaZloa, sxvadasxvac iyos. Tu, vTqvaT, gvaqvs raime simravle A, da

viciT, rom is oTxi a,b,c da d elementisagan Sedgeba, maSin, miRebulia,

rom misi CamonaTvalis saxiT Cawera Semdegnairad moxdes: A = {a,b,c,d}.

xolo Canaweri {x: x∈A} ikiTxeba rogorc “im (an, yvela im) elementTa

simravle, romelic ekuTvnis A-s” da aRniSnavs im viTarebas, rodesac simravle gasazRvrulia garkveuli wesiT. Cven SemTxvevaSi, es wesi

elementis A-sadmi kuTvnlebas gulisxmobs. sxva sityvebiT, x SeiZleba

iyos an a, an b, an c, an d. 1.3. zogjer xdeba, rom simravleTa ganxilvisas iseT viTarebas vawydebiT,

rodesac erTi simravle mTlianad Seicavs meore simravlis

elementebs, anda es elementebi erTi da igivea. magaliTad, luwi

ricxvebis simravle Seicavs 20-ze naklebi luwi ricxvebis simravles,

xolo im ricxvTa simravle, romlis 2-ze gayofaa SesaZlebeli, mas

mTlianad emTxveva. rodesac adgili aqvs an erT, an meore SemTxvevas,

vambobT, rom meore simravle pirvelis qvesimravles warmoadgens. amis

Cawera Semdegnairad xdeba: vTqvaT, Cveni simravleebia A da B. Canaweri

A⊆B, an B⊇A niSnavs, rom A qvesimravlea B-si. rodesac aseTi ram ar xdeba, vxmarobT niSans ⊈ an ⊉ Sesabamisad. rodesac xazi gvinda gavusvaT im garemoebas, rom raime simravle, vTqvaT, igive A, aris

meore simravlis (vTqvaT, B-s) qvesimravle, Tumca am ukanasknels ar

emTxveva, vxmarobT Semdegi saxis Canawers A⊂B, an B⊃A da vambobT, rom

A sakuTrivi qvesimravlea B –si. igives Cawera, sxvaTaSoris,

sxvanairadac SeiZleba: A⊆B igivea, rac {x: Tu x∈A, maSin x∈B}. 1.4. davubrundeT Cvens saklaso oTaxs. vTqvaT, am momentisaTvis masSi arc

erTi studenti ar imyofeba. maSin ras unda niSnavdes gamoTqma am

klasSi amJamad aq myofi studentebis simravle? ra Tqma unda,

sicarieles. sxva sityvebiT, simravle, romelic elementebs ar Seicavs

(anu ar arsebobs iseTi sagani, romelic misi elementi iqneboda) aris

carieli. carieli simravlis aRsaniSnavad, Cveulebriv, xmaroben ∅ simbolos. maSasadame, Tu A carieli simravlea, es aseTnairad

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SegviZlia CaweroT: A=∅, anda A-Tvis vixmaroT Canaweri: {x: x∈A da x∉A}. sakiTxi, Tu ratom gvWirdeba aseT simravleebze laparaki, da xom ar

jobia, saerTod ugulvebelvyoT isini, sinamdvileSi arc ise martivia.

amitom aq masze ar SevCerdebiT. SevTanxmdeT, rom cariel

simravlesac iseTive ufleba aqvs iyos msjelobisa Tu gansjis

obieqti, rogorc maTematikaSi 0-s. sxvaTa Soris, maTi Tvisebebic

saocar msgavsebas amJRavneben.

1.5. garda carieli simravlisa, miRebulia visaubroT aseve ori saxis

metad Tavisebur simrvleebze. esenia universaluri simravle da

erTeulovani simravleebi. pirvel simravleze laparakisas yuradReba

unda mivaqcioT im garemoebas, Tu rogor aris SemosazRvruli are im

obieqtebisa, romelic Cveni msjelobis sagnadaa qceuli. zogadad, es

sxvadasxva velebia, da gansxvaveba, xSirad, friad mniSvnelovan

xasiaTs atarebs. zemoT motanil magaliTSi Cveni yuradRebis obieqti

konkretuli saklaso oTaxi iyo, romlis gareT Cven ar gavdiodiT da

sxvadasxva simravleebsac mis SigniT varCevdiT. am SemTxvevaSi,

universaluri simravlis rols (mas zogjer bWobis universumsac

eZaxian) am klasSi arsebuli yvela sagnis simravle asrulebs. sxva

SemTxvevaSi, anu rodesac Cveni gansja sxva obieqtebzea mimarTuli,

vTqvaT, igive planetebze, anda naturalur ricxvebze, bWobis

universumic da universaluri simravlec Sesabamisad ciur sxeulTa

simravle da ricxvTa RerZi SeiZleba rom iyos. vambobT, rom

SeiZleba, vinaidan, maT garda, sxvagvari universumebicaa SesaZlebeli.

es imazea damokidebuli, Tu ra doneze virCevT im sazRvrebs, romlis

iqiT msjelobisas gasvlas ar vapirebT. magaliTad, igive planetebTan

dakavSirebiT, es universumi, sul mcire, Mmzis sistema SeiZleba rom

iyos, anda, sulac, samyaroSi arsebul galaqtikaTa erToblioba.

naturaluri ricxvebisTvis ki mTeli ricxvebis, anda, sulac namdvili

ricxvebis simravle. sxvaTa Soris, amis gamo, sasargebloa, rom

simravleebiT operirebisas, kargad gvqondes gansazRvruli is

universumi, romlis qvesimravleebsac Cveni gansaxilveli simravleebi

warmoadgens. universaluri simravlis aRsaniSnavad miRebulia

vixmaroT 𝐔𝐔 (pirveli aso sityvisa UNIVERSAL). rac Seexeba erTeulovan

simravleebs, anu simravleebs, romlebic mxolod erT elements

Seicavs, SesaZloa, gaazrebisas igive sirTuleebi axldes, rac

carieli simravlis gaazrebas. SevTanxmdeT, rom amjerad es

sirTuleebi ar gvainteresebs, xolo erTelementiani simravleebis

SenarCuneba iseve sasargebloa (ufro, albaT, mosaxerxebeli) rogorc

carielisa.

1.6. venis diagramebi. 1881 wels britaneTSi gamoica britaneli

logikosisa da filososfosis jon venis (1834-19230) wigni saTauriT

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simboluri logika (Symbolic Logic), sadac, simravleebze saubrisas man sistemuri saxiT gamoiyena e.w. diagramebis meTodi, romelic Zalian

male popularuli gaxda mecnierTa Soris. dRes mas warmatebiT

iyeneben simravleTa TeoriaSi, logikaSi, statistikasa da

kompiuterul mecnierebebSi. amis gamo, am meTods, mis sapativcemulod,

venis diagramebi ewoda. meTodis arsi Zalian martivia.

TvalsaCinoebisTvis, simravleebi wreebiTaa warmodgenili, romlebic

zogjer kveTen, zogjer faraven an nawilobriv faraven erTmaneTs.

mgaliTad. Tu gvaqvs ori, A da B simrvle, maSin is, rom pirveli

meores sakuTrivi qvesimravlea, venis diagramiT Semdegnairad

gamoixateba:

A⊂B

qvemoT, venis diagramebs xSirad mivmarTavT simravleTa sxvadasxva

Tvisebebisa da maT Soris arsebuli mimarTebebis aRsawerad.

$2. ZiriTadi damokidebulebebi simravleebs Soris.

is, rom simravleebi unda gansxvavdebodnen im obieqtebisagan, romlebic maT

elementebs warmoadgens, TiTqos TavisTavad gasagebi da cxadi ram unda

iyos. marTlac am klasSi myofi skamebis cimravle srulebiTac ar aris

skami, xolo am klasSi myofi studentebis simravle ar aris studenti.

Tumca es yovelTvis ase araa, raSic advilad davrwmundebiT, Tu ki iseT

erTobliobaze daviwyebT msjelobas, rogorsac, magaliTad, yvela

simravlis simravle qmnis. Tumca amaze cota qvemoT. amJamad, Cven sul sxva

problema gvainteresebs. cxadia, rom simravleebi Tavad SeiZleba qmnidnen

simravleebs. magaliTad, varskvlavTa sistemebi, romlebic ciur sxeulTa

garkveul simravleebs qmnian, ufro didi sistemebis, galaqtikebis

elementebad SegviZlia moviazroT. es ukanasknelni, aseve qmnian garkveul

erTobliobas da a.S. ufro TvalsaCino magaliTebis moZiebacaa

SesaZlebeli, Tu ki Cvens universumad ricxvTa RerZs moviazrebT. im

B

A

12 uamravi SesaZleblobis ganxilvidan, rasac aRniSnuli universumi sxvadasva

simravleTa asagebad iZleva, SegviZlia erT saintereso garemoebas

mivaqcioT yuradReba: rogori ori aracarieli simravlec ar unda aviRoT,

maT Soris mxolod xuTi damokidebulebaa SesaZlebeli. kerZod,

simravleebi Seileba emTxveodnen erTmaneTs, SesaZloa isini nawilobriv

emTxveodnen erTmaneTs, SeiZleba rom erTi iyos meores nawili, anda

SesaZloa es damokidebuleba piriqiT xorcieldebodes, da bolos,

simravleebi saerTod ar “exebodnen” erTmaneTs, sxva sityvebiT, ar

arsebobdes mocemul universumSi isTi elementi, romelic maT, SesaZloa,

saerTo qondeT. ganvixiloT es viTarebebi cal-calke. vTqvaT Cveni

universumia 𝐔𝐔 da masSi nebismierad SevarCieT ori aracarieli simravle A

da B:

2.1. ras unda niSnavdes, rom isini erTmaneTs emTxveva? rogorca Cans,

sxvas arafers, garda imisa, rom orive simravle erTi da igive

elementebisaganaa Sedgenili. sxva sitvebiT. Tu x∈ A, maSin x∈ B, da

piriqiTac, Tu raime elementi, vTqvaT y ekuTvnis B-s ( y∈ B), maSin y

ekuTvnis A-s ( y∈ A). sxvaTa Soris, es viTareba erTi winadadebiTac SegviZlia gamovTqvaT: nebismieri elementisaTvis 𝐔𝐔-dan, Tu iqidan, rom

is ekuTvnis A-s, gamodis, rom is amavdroulad ekuTvnis B-s, da piriqiT, maSin es simravleebi igveobrivi simravleebi arian. venis diagramaze es

SeiZleba ase aRibeWdos:

simravleTa magaliTebi, romlebic erTmaneTs emTxveva im azriT, rom

erTi da igive elementebisgan Sedgebian, mravlad SegviZlia movitanoT

𝐔𝐔

A=B

13

da es savarjiSos saxiT SeiZleba gakeTdes. aq yuradReba mivaqcioT

erT saintereso garemoebas, romelic Cveni ganmartebidan gamodis:

yvela carieli simravle erTmaneTis igiveobrivia (ratom?), rasac ver

vityviT, Tundac, erTelementian simravleebze (ratom?).

2.2. ras unda niSnavdes is garemoeba, rom mocemuli simravleebi

nawilobriv emTxveva erTmaneTs? aqac saintereso viTareba gvaqvs.

nawilobriv damTxveva niSnavs rom A da B simravleebs, romelTagan arc

erTi ar aris meoris qvesimravle, raRac elementebi (Tundac erTi)

saerTo aqvT. sxva sityvebiT, universumSi unda arsebobdes erTi

elementi mainc iseTi, rom is ekuTvnis rogorc erT, ise meore

simravlesac. bunebrivia, aseTi elementi metic SeiZleba rom iyos.

nawilobriv emxTveva erTmaneTs, magaliTad, mamakacebisa da samxedro

mosamsaxureTa, 20 wlamde vaJebisa da studenti vaJebis, poetebisa da

momRerlebis simravleebi. maSasadame, piroba imisa, rom A da B

simravleebi nawilobriv emTxveva erTmaneTs, aris is, rom A⊈B da B⊈A da, amave dros, universumSi arsebs iseTi obieqti (obieqtebi), vTqvaT a,

romlisTvisac samarTliania rogorc a∈ A, aseve a∈ B. venis diagramiT es SeiZleba ase gamoisaxos:

mocemul diagramaze A da B simravleebis urTierTgadamfaravi

(gadamkveTi) nawili karg warmodgenas gviqmnis imaze, Tu ras niSnavs rom

isini erTmaneTs nawilobriv emTxveva.

B

A

14 2.3. axla vnaxoT Tu rasTan gvaqvs saqme, rodesac erTi simravle meoris

nawilia: es is SemTxvevaa, rodesac erTi simravle meoris sakuTriv

qvesimravles warmoadgens. sxva sityvebiT, misi yoveli elementi amave

dros meoris elementicaa, xolo meores damatebiT, sxva elementebic

gaaCnia. wina paragrafSi motanili diagrama swored am SemTxvevas

asaxavs. sxvaTa Soris, aqedan erTi, metad saintereso, garemoeba gamodis:

carieli simravle yvela sxva simravlis qvesimravles warmoadgens

(ratom?).

2.4. ra viTarebasTan gvaqvs saqme, rodesac aracarieli simravleebi, ase

vTqvaT, erTmaneTs ar “exebian”? aseT simravleebs gamijnul

simravleebs uwodeben. bunebrivia, vifiqroT, rom A da B simravleebi

gamijnulebi mxolod maSin iqnebian, rodesac maT arc erTi saerTo

elementi ar gaaCniaT. aseTi simravleebis magaliTebis moZieba sakmaod

advilia. universumad, magaliTad rom Cveni saklaso oTaxi aviRoT, am

klasSi myofi skamebis cimravle da am klasSi myofi studentebis

simravle sworedac rom gamijnulebi unda iyos, vinaidan arc erTi

studenti ar aris skami da arc erTi skami, Tavis mxriv, ar aris

studenti. anu, saqme gvaqvs Semdeg viTarebasTan: nebismieri

elementisaTvis 𝐔𝐔-dan, Tu iqidan, rom is ekuTvnis A-s, gamodis, rom is

ar ekuTvnis B-s, da piriqiT, Tu ki is ekuTvnis B-s, maSin is ar

ekuTvnis A-s, xolo A da B aracarieli simravleebia 𝐔𝐔-dan, maSin A da B gamijnuli simravleebia. venis diagramaze es SeiZleba ase aRibeWdos:

N

yuradReba mivaqcioT im garemoebas, rom aracariel simravleebs Soris sxva

aranairi damokidebuleba ar arsebobs, Tumca, erT-erTi aucileblad

xorcieldeba. savarjiSos saxiT, SegviZlia davasabuToT isic, rom Tu

adgili aqvs erT-erTs, maSin sxva arc erTi ganxorcielebuli ar iqneba.

dasabuTebisaTvis SeiZleba venis diagramebi gamoviyenoT.

A B

15 $3. operaciebi simravleebze:

cota xniT daviviwyoT, rom simravleebi nebismier elementTa erTobliobebs

warmoadgens da maT rogorc garkveuli tipis maTematikur obieqtebs, ise

SevxedoT. MmaTematikuri obieqtebis erT-erT mniSnelovan Taviseburebas

warmoadgens is, rom maTze garkveuli operaciebis Catarebaa SesaZlebeli.

magaliTad, naturluri ricxvebisaTvis esenia Sekreba, gamokleba da

gamravleba. gayofas, rogorc cnobilia, naturalur ricxvTa simravles

gareT gavyavarT. vnaxoT, ra operaciebia SesaZlebeli, Tu ricxvebis

nacvlad Tavad simravleebi gvaqvs? Tavdapirvelad, CamovyalibdeT, Tu ras

davarqvaT operacia simravleebze? Sinaarsobrivad es iseT moqmedebas

gulisxmobs, romelic ori, an meti simravlidan savsebiT konkretul, axal

simravles gvaZlevs (anu migviTiTebs im wesze, Tu rogor unda aigos es

axali simravle, rodesac raRac mocemuloba gagavaCnia). SedarebisTvis,

gavixsenoT, Tu rogor ganvsazRvravdiT ariTmetikul moqmedebebs

maTematikis saskolo kursSi. sxvaTaSoris, simravleebze

gansaxorcielebeli operaciebi Zalian hgavs kidec ariTmetikul

moqmedebebs im azriTac, rom wesebi, romelic ZalaSia ariTmetikuli

moqmedebisTvis, aseve ZalaSia am operaciebisTvisac.

3.1. gaerTianeba da misi Tvisebi. ori A da B simravlis gaerTianebad

miCneulia iseTi simravle, romelic Seicavs yvela elements A- dan da yvela

elements B-dan. sxva sityvebiT, A da B simravlis gaerTianeba agebulia A da

B simravlis elementebis TavmoyriT, SeerTebiT.

magaliTad, luwi da kenti naturaluri ricxvebis simravleTa gaerTianeba

aris naturalur ricxvTa simravle.

simravleTa gaerTianebis operaciis aRsaniSnavad ixmareba simbolo ∪,

romelic mocemul simravleTa Soris iwereba. ase, rom Canaweri A∪B niSnavs

A da B simravleebis gaerTianebas.

3.1.1. vTqvaT,AA=B. ras unda warmoadgendes maSin AA∪B? davuSvaT, gvaqvs

A=B={a,b,c,d}, anu, A= {a,b,c,d}, da B={a,b,c,d}. Tu Cvens gamartebas moviSveliebT, advilad davrwmundebiT, rom AA∪B=A anda, rac igivea,

A∪B=B. TvalsaCinoebisaTvis mivmarToT venis diagramas:

A=B

16

erT-erTi Sedegi, sxvaTa Soris, aris is, rom ∅∪∅=∅ .

3.1.2. vTqvaT,AA⊂B. ras unda warmoadgendes maSin AA∪B? davuSvaT A= {a, c },

xolo B={a,b,c,d}. gaerTianebis gamarteba moviSvelioT da vnaxavT, rom am

SemTxvevaSi, B, ase vTqvaT, “STanTqavs” A-s, anu, rac igivea, rom AA∪B=B. TvalsaCinoebisaTvis mivmarToT venis diagramas:

es arc unda iyos gasakviri. magaliTad, luwi da naturaluri ricxvebis

simravleTa gaerTianeba xom isev naturalur ricxvTa simravlea.

3.1.3. vTqvaT,AA⊃B. ras unda warmoadgendes A∪B? pasuxs advilad mivagnebT,

Tu ki wina SemTxvevaSi A da B simravleebs adgilebs SevunacvlebT.

1.4. vTqvaT,AA da B nawilobriv emTxveva erTmaneTs. ras unda warmoadgendes

aseT SemTxvevaSiAA∪B? davuSvaT A= {a,b, c }, xolo B={c,d,j}. gamartebis

Tanaxmad es unda iyos {a,b, c,d,j }, sxva sityvebiT, es aris simravle, romelic

maTi saerTo elementebis garda Seicavs maT im elementebac, romelic maT

gansxvavebuli gaaCniaT. savarjiSos saxiT aageT Sesabamisi venis diagrama.

3.1.5. vTqvaT, AA da B gamijnuli simravleebia. ras unda warmoadgendes A

A∪B? ganmartebis mixedviT, es A da B simravlis elementebis meqanikuri

SeerTebiT miRebuli simravle iqneba.

3.1.6. vTqvaT, AA carieli simravlea. advili sanaxavia, rom misi gaerTianeba

nebismier sxva aracariel simravlesTan Tavad am simravlis igiveobrivi

iqneba.

3.1.7. TuAA universaluri simravlea, aseve advili sanaxavia, rom misi

gaerTianeba sxva nebismier simravlesTan isev A iqneba (axseniT 3.1.6 da 3.1.7)

B A

17 3.1.8. venis diagramebis gamoyenebiTa da gaerTianebis ganmartebaze

dayrdnobiT, savarjiSos saxiT, avxsnaT gaerTianebis Semdegi ZiriTadi

Tvisebebi:

A⊆A∪B

A⊆A∪A

A∪B⊆B∪A

B∪A⊆A∪B

((A∪B)∪C))⊆(A∪(B∪C))⊆(A ∪B∪C)

3.2. kveTa. zemoT Cven erTmaneTis nawilobriv gadamfarav simravleebze

visaubreT. xSirad, aseT simravleebs urTierTgadamkveT simravleebs, xolo

Tavad gadafarvas, kveTas uwodben. kveTa operciadac SegviZlia

warmovidginoT, romlis saSualebiT, ori an meti simravlidan axali

simravle SegviZlia gamovyoT, romelic xasiaTdeba imiT, rom Sedgeba im da

mxolod im elementebisagan, romelis mocemul simravleebs saerTo aqvT.

kveTas ∩ simboloTi aRniSnaven. sxvanairad, es SeiZleba Semdegnairadac

gamovTqvaT: x∈(A∩B) zustad maSin,rodesac x∈A da x∈B, anda, sxva sityvebiT,

x∈A da x∈B, sadac A da B aracarieli simravleebia 𝐔𝐔-dan, maSin da mxolod

maSin, rodesac x∈(A∩B). Tu kargad davakvirdebiT qvemoT motanil

diagramas, sadac sami simravlis SemTxvevaa mocemuli, kargad davinaxavT,

Tu ra igulisxmeba kveTaSi.

18 sxvaTa Soris, am diagramidan kargad Cans isic, Tu ra unda iyos kveTa, Tu

ki simravleTa ZiriTad damokidebulebebs davubrundebiT da im SemTxvevebs

vnaxavT, rodesac simravleebi erTmaneTs nawilobriv ar faraven:

3.2.1. Tu AA⊂B, maSin AA∩B=A

3.2.2. Tu AA⊃B, maSin AA∩B=B

3.2.3. xolo, rodesac AA da B gamijnuli (aracarieli) simravleebia,

bunebrivia, maTi kveTa carieli simravle iqneba. amitom, is rom simravleebi

gamijnulia, SesaZloa Semdegnairadac CavweroT: A∩B=∅, sadac A da B aracarieli simravleebia 𝐔𝐔-dan.

3.2.4 vTqvaT, AA carieli simravlea. advili sanaxavia, rom misi kveTa

nebismier sxva aracariel simravlesTan isev carieli simravle iqneba.

3.2.5. TuAA universaluri simravlea, aseve advili sanaxavia, rom misi kveTa

sxva nebismier simravlesTan am ukanasknelis toli iqneba(axseniT 3.2.4 da 3.2.5)

3.1.8. venis diagramebis gamoyenebiTa da kveTis ganmartebaze dayrdnobiT

savarjiSos saxiT, avxsnaT kveTis Semdegi ZiriTadi Tvisebebi:

A∩B⊆A

A∩B⊆B

A∩B⊆B∩A

((A∩B) ∩C))⊆(A∩(B∩C))⊆(A ∩B∩C)

3.3. sxvaoba: simravleTa sxvaobad, Cveulebriv, miCneulia iseTi simravle,

romelic agebulia saklebi simravlis im da mxolod im elementebisagan,

romlebic ar ekuTvnis maklebs. sxvaobisTvis xSirad xmaroben simbolos

“_”’ rogorc es maTematikaSia miRebuli. zogjer, sxva simbolosac xmaroben.

vnaxoT, ra unda iyos ori simravlis sxvaoba, maTi ZiriTadi

damokidebulebebis realizaciis SemTxvevaSi:

3.3.1. vTqvaT,AA=B. ras unda warmoadgendes maSin AA-B? TvalsaCinoebisaTvis mivmarToT venis diagramas:

19

vinaidan, AA-Si ar moiZebneba elementi, romelic

ar ekuTvnis B-s, amitom es sxvaoba carieli

simravle unda iyos, rac, rogorca Cans,

bunebrivicaa.

3.3.2. vTqvaT,AA⊂B. ras unda warmoadgendes maSin AA-B? sxvaobis gamarteba

moviSvelioT da vnaxavT, rom am SemTxvevaSi, vinaidan B “STanTqavs” A-s,

sxva sityvebiT, A-Si ar moiZebneba elementi, romelic, amavdroulad, B-s elemeti ar aris, amitom AA-B=∅. TvalsaCinoebisaTvis mivmarToT venis

diagramas:

A A-B=∅

3.3.3. vTqvaT,AA⊃B. ras unda warmoadgendes A-B? pasuxs advilad mivagnebT,

Tu ki wina SemTxvevaSi A da B–s adgilebs SevunacvlebT.

am SemTxvevaSi, “mSTanTqmeli” simravle

aris A, da rogorc aseTs, aucileblad

gaaCnia iseTi elementebi, romlebic ar

gaaCnia B–s. Sesabamis venis diagramaze A simravlis wiTlad moniSnuli nawili

swored (A-B)-s warmoadgens. sxvaTa

Soris, am diagramaze isic Cans, rom am

SemTxvevaSi, (A-B)=A-(A∩B).

A=B

B

A

B

A

20

3.3.4. vTqvaT,AA da B nawilobriv emTxveva erTmaneTs. ras unda warmoadgendes

aseT SemTxvevaSiAA-B? davuSvaT A= {a,b, c }, xolo B={c,d,j}. gamartebis

Tanaxmad es unda iyos {a,b}, sxva sityvebiT, es aris simravle, romelic A-s mxolod im elementebs Seicavs, romelic ar ekuTvnis maT kveTas. anu,

A-B=A-(A∩B).

venis diagramaze A simravlis wiTlad

moniSnuli nawili swored (A-B)-s warmoadgens.

3.3.5. vTqvaT, AA da B gamijnuli simravleebia. ras unda warmoadgendes A

A-B? ganmartebis mixedviT, es aris A .

3.3.6. vTqvaT, AA carieli simravlea. advili sanaxavia, rom misi da sxva

nebismier simravlis sxvaoba isev carieli simralea: A-B=A

3.3.7. TuAA universaluri simravlea, maSin misi sxvaoba nebismier sxva

simravlesTan warmoadgens mTel danarCen universums am simravlis

elementebis garda.

A B

𝐔𝐔

B

21 Sesabamis venis diagramaze es sxvaoba molurjo feriTaa moniSnuli. sxvaTa

Soris, mas gansakuTrebuli mniSvneloba eniWeba da amis gamo, specialuri

terminiTa da simboloTi aRniSnaven. universaluri da mocemuli simravlis

sxvaobas am simravlis damateba ewodeba da mas StrixiT gamoxataven.

magaliTad, Tu B mocemuli simravlea, maSin misi damateba iqneba B′. sxva sityvebiT: U-B=B′. aqedan, sxvaTa Soris, gamodis isic, rom U-B′=B da

B′∪B=U, xolo B′∩B=∅.

3.4. simetriuli sxvaoba. simravleTa TeoriaSi ganixilaven sxvaobis kidev

erT, metad saintereso saxeobas, e.w. simetriuli sxvaoba. ras warmoadgens

simravle, romelic am operaciis Sedgad aris miRebuli? iTvleba, rom AA

da B simravleTa simetriuli sxvaoba aris iseTi simravle, romelic

Seicavs im da mxolod im elementebs, romlebic ekuTvnis mxolod AA-s,

anda mxolod B-s. magaliTad, Tu gvaqvs, rom A= {a,b, c }, da B={c,d,j}, maSin maTi simetruli sxvaoba iqneba simravle {a,b, d,j}. vnaxoT, ra suraTi gveqneba, Sesabamis venis diagramas Tu mivmarTaT da simravleTa Soris

ZiriTad mimarTebebs gaviTvaliswinebT. simetriuli sxvaobisaTvis vixmaroT

niSani “/”.

3.4.1. A=B

vinaidan, AA-Si ar moiZebneba elementi, romelic

ar ekuTvnis B-s, da piriqiT, amitom es sxvaoba iqneba ∅.

3.4.2. A⊂B.

ganmarteba moviSvelioT da vnaxavT, rom am

SemTxvevaSi, vinaidan B “STanTqavs” A-s, sxva

sityvebiT, A-Si ar moiZebneba elementi,

romelic, amavdroulad, B-s elemeti ar aris,

amitom es sxvaoba iqneba B-s mxolod im

elementTa simravle, romelic ar gaaCnia A-s,

A=B

B

A

22

sxva sityvebiT, is daemTxveva (B-A)-s, rac Sesabamis diagramaze wiTliTaa moniSnuli.

3.4.3. A⊃B.

am SemTxvevaSi, “mSTanTqmeli” simravle aris

A, amitom simetriuli sxvaoba igivea, rac (A-B).

3.4.4. AA∩ B ≠∅.

venis diagramaze A da B simravleTa

wiTlad moniSnuli nawili swored

saZiebel simravles warmoadgens.

3.4.5. AA∩ B =∅.

venis diagramaze A da B simravleTa

wiTlad moniSvna imaze mianiSnebs,

rom am SemTxvevaSi, ganmartebidan

gamomdinare, saZiebeli simravle

mocemul simravleTa gaerTianebas

warmoadgens.

A

B

A B B

A B

23

3.4.6. vTqvaT, AA carieli simravlea. advili sanaxavia, rom nebismieri

aracarieli B-sTvis samarTliania Semdegi tolobebi:

• A/A=∅ • A/B=B

• B/A=B

axseniT, ratom?

3.4.7. TuAA universaluri simravlea, maSin misi simetruli sxvaoba

nebismier sxva simravlesTan warmoadgens mTel danarCen universums am

simravlis elementebis garda. sxvaTa Soris, es 3.4.2 da 3.4.3.-dan

gamomdinareobs. amgvarad, SegviZlia vTqvaT, rom simetruli sxvaoba, zog

SemTxvevaSi, emTxveva sxvaobas, Tumca es ori sxvadasxva operaciaa.

3.5. simravleTa (kartezianuli) namravli. simravleebisaTvis kidev erTi

operacia SegviZlia ganvsazRvroT, e.w. maTi kartezianuli namravli. Tu

mocemuli gvaqvs ori A da B simravle, aRniSnuli namravli iseTi

simravlea, romelic Sedgenilia wyvilebisagan romelTa pirveli wevri A simravlis elementia, xolo meore wevri - B simravlisa. namravlSi yvela

aseTi wyvilia mocemuli. magaliTad, Tu A aris {a₁,a₂,a₃}, xolo B aris {b₁,b₂,b₃}, maSin kartezianuli namravli iqneba wyvilebis Semdegi simravle:

{‹a₁,b₁›,‹a₁,b₂›,‹a₁,b₃›,‹a₂,b₁›,‹a₂,b₂›,‹a₂,b₃›,‹a₃,b₁›,‹a₃,b₂›,‹a₃,b₃›}. simravleebis

kartezianul namravls X-iT aRniSnaven.

§4. usasrulo simravleebi da maTTan dakavSirebuli zogierTi siZnele.

raselis paradoqsi.

pirvel paragrafSi, simravlebis aRwerisas, Cven usasrulo simravleebic

vaxseneT. is, rom simravlis usasruloba gonebisaTvis ucxo ar unda iyos

da misi moazreba, garkveulwilad, bunebrivic ki Cans, TiTqo sababsac

iZleva imisaTvis, rom maTze iseve, rogorc Cveulebriv, sasrul

simravleebze vilaparakoT. praqtikulad, Cven asec viqceviT. magaliTad,

winadadebebSi

“samyaroSi varskavlavTa raodenoba usasruloa” da

“yvela naturaluri ricxvi aris mTeli da dadebiTi ricxvi”,

romlebsac Cveulebriv warmovTqvamT, da metic, WeSmaritadac miviCnevT,

igulisxmeba, rom aRniSnuli usasruloba gonebisTvis iseTive mocemulobaa,

24 rogorc “am saklaso oTaxSi magidebisa da skamebis simravle” . es rom ase ar iyos, maSin maTi WeSmaritebadac miCneva gagviZneldeboda. arada, Tu ki

pirvelze, vTqvaT, rame filosofiur mosazrebaTa gamo, vinmem SeiZleba eWvi

gamoTqvas, meore winadadeba ueWveladaa miCneuli. miuxedavad amisa,

usasrulo simravleebi metad specifikuri simravleebia da maT garkveuli

“ucnaurobebi” axasiaTebs. erT-erTi aseTi “ucnaurobaa”, magaliTad, is, rom

raodenobrivi TvalsazrisiT, usasrulo simravleebi Tavisive sakuTrivi

qvesimravleebis tolZalovania. rodesac vambobT “tolZalovani”,

vgulisxmobT, rom mocemul simravleebs elementTa erTi da igive

raodenoba gaaCnia, rac, sxvaTa Soris, srulebiTac ar niSnavs maT

igiveobas, miuxedavad imisa, rom igiveobrivi simravleebi tolZalovania.

magaliTad, tolZalovania naturalur da luw ricxvTa simravleebi, Tumca

ver vityviT, rom isini igiveobrivia. sasruli simravleebis tolZalovnebis

saCveneblad, xSirad Semdeg proceduras mimarTaven: cdiloben erTi

simravlis elementebi SeuTanadon (ase vTqvaT, “miaban”) meore simravlis

elementebs iseTnairad, rom TiToeul elements mxolod erTi elementi

eTanadebodes. Tu aseTi “mibmis” Sedegad, erTi simravlis yvela elements

aRmoaCnda Tavisi “wyvili” meore simravleSi da piriqiT, maSin vityviT, rom

isini tolZalovani yofilan. SesaZloa ise aRmoCndes, rom erTi simravlis

yvela elements yavs Tavisi “mewyvile” meore simravleSi, xolo meore

simravlis elementebs Soris aris iseTebic, romlebic Sewyvilebis gareSea

darCenili, maSin es simravleebi ar iqneba tolZalovani. sxvaTa Soris, am

SemTxvevaSi amboben, rom meore simravlis simZlavre metia pirvelisaze, rac

sxvas arafers niSnavs, garda imisa, rom meore simravleSi ufro meti

elemetia vidre pirvelSi. magaliTad, Tu Cvens saklaso oTaxs gavixsenebT,

SegviZlia warovidginoT, rom masSi magidebis raodenoba naklebia skamebis

raodenobaze, xolo es ukanaskneli tolia klasSi amJamad myofi

studentebis raodenobisa. maSin SegviZlia vTqvaT, rom am klasSi myofi

studentebis da am klasSi myofi skamebis simravleebi tolZalovania,

xolo am klasSi myofi skamebis da am klasSi myofi magidebis simravleebi

tolZalovani araa da pirveli, ase vTqvaT, metia meoreze. am magaliTidan,

sxvaTa Soris, kargad Cans isic, rom SesaZloa sasrili simravleebi

tolZalovani ar iyon, magram es ar niSnavs imas, rom romelime meoris

qvesimravlea. imaSi, rom usasrulo simravleebi Tavisive sakuTrivi

qvesimravleebis tolZalovania, advilad davrwmundebiT, Tu ki naturalur

ricxvTa simravles luwi ricxvebis simravles SevadarebT. sxva sityvebiT,

es niSnavs, rom usasrulo simravleebisaTvis, mTeli misi nawilis tolia.

metic, Tu usasrulo simravles mis usasrulo qvesimravles gamovaklebT,

Sedegad, zogjer, isev usasrulo simravles viRebT. Cvens magaliTSi,

naturalur da luw ricxvTa simravleebis sxvaoba kenti ricxvebis

simravles gvaZlevs, romelic iseve usasruloa, rogorc luwi ricxvebis

25 simravle. aseve “ucnaurobad” SegviZlia miviCnioT is garemoebac, rom Tu

usasrulo simravles raime elements (elementebs) davamatebT, Sedegad isev

usasrulo simravle gveqneba. es, faqtiurad, niSnavs imas, rom simravle,

romelic Tundac erTi elementiT mets Seicavs, tolia simravlisa,

romelsac es elementi aklia, rac savsebiT gamoricxulia, Tu ki saqme

sasrul simravleebTan gveqneboda.

is, rom simravleTa usasruloba garkveul sirTuleebs Seicavda,

maTematikosebisaTvis imTaviTve cxadi Seiqna, rogorc ki maT am

simravleebiT operireba daiwyes da simravleTa Teoriis maTematikis

dasafuZneblad gamoyeneba cades. metic, sul male aRmoCenil iqna rigi

siZneleebisa, romelTa gadalaxva, erTi SexedviT, SeuZlebelic Canda da

romlebmac maTematikis dafuZnebis yvela imdroindeli cda Rrma krizisSi

moaqcia. es siZneleebi simravleTa Teoriis paradoqsebis (an antinomiebis)

saxeliTaa cnobili da me-20 saikunis dasawyisis maTematikosTa da

logikosTa gansakuTrebuli yuradRebis obieqts warmoadgenda. paradoqsi,

an antinomia iseT viTarebas hqvia, rodesac savsebiT misaRebi

debulebebidan urTierT gamomricxav daskvnebamde (winaaRmdegobamde)

mivdivarT. yvelaze cnobili am paradoqsebidan e.w. raselis paradoqsia,

romelic man 1901 wlis gazafxulze aRmoaCina, rodesac Tavis cnobil

wignze Principles of Mathematics (maTematikis principebi) (1903), muSaobda. paradoqsi Cndeba iseTi simravleebis ganxilvisas, romlebic Tavis Tavs

elementad ar Seicavs. adre Cven gakvriT SevexeT aseT simravleebs,

rodesac aRvniSneT, rom am saklaso oTaxSi myofi magidebis simravle

Tavad magida ar aris. metic, simravleTa umravlesoba, romelTanac Cven

saqme gvaqvs, swored rom aseTi simravleebia. magaliTad, ricxvTa

nebismieri simravle ar aris ricxvi, xolo varskvlavebis simravle ar aris

varskvlavi. ase, rom savsebiT bunebrivi Cans ganvixiloT yvela aseTi

simravlis simravle, anu simravle yvela iseTi simravlisa (vTqvaT, K),

romlebic Tavis Tavis elementi araa. axla davsvaT kiTxva, aris Tu ara K

Tavis Tavis elementi? pasuxi gamaognebelia: Tu K∈K, maSin K∉K, xolo Tu

K∉K, maSin K∈K. raselis paradoqsis ramodenime saxesxvaoba arsebobs,

romelTagan, zogierTi, sakmaod saxalisoa. magaliTad, soflis dalaqis

paradoqsi, romelic SemdegSi mdgomareobs: dadgenilia, rom soflis

dalaqi emsaxureba soflis mxolod im macxovreblebs romlebic TviTon ar

iparsaven wvers da mis garda aravin aravis ar parsavs. vin unda gaparsos

Tavad dalaqi? unda iTqvas, rom Tavis droze, simravlisTeoriuli

paradoqsebis aRmoCenam mniSvnelovano roli Seasrula maTematikisa da

logikis ganviTarebaSi. Tumca, mets aq amaze ar SevCerdebiT: es metad

specifikuri, sainterso da Rrma Temaa.

26 Tavi II. propoziciuli logikis sawyisi cnebebi da ZiriTadi

kanonebi

$1. ras vuwodebT propoziciul logikas? martivi da rTuli winadadebebi.

sawinadadebo makavSireblebi.

propoziciur, anu winadadebaTa logikas, vuwodebT logikis im nawils,

romelic Seiswavlis winadadebTa Soris arsebul damokidebulebebs,

romelnic, Tavis mxriv, mxolod da mxolod, am winadadebaTa agebis

TaviseburebebiTaa gapirobebuli. avxsnaT es viTareba kargad: winadadebaTa

mTeli erToblioba, romelic Cveni ganxilvis obieqtad SegviZlia

moviazroT (aseTs ki, ZiriTadad, raime enis, vTqvaT, igive qarTulis,

TxrobiTi winadadebebis erToblioba Seadgens), pirobiTad, SegviZlia or

nawilad gavyoT. erT nawils is winadadebebi Seadgens, romelic, ase

vTqvaT, martivia, im azriT, rom maTi danawevreba Semadgenel

winadadebebad SeuZlebelia, vinaidan isini amdagvar nawilebs ar Seicavs.

magaliTad, qarTul enaSi aseTi iqneboda

me vxedav mzes, anda

TbilisSi aqtiuri politikuri cxovrebaa, anda

kavkasia mniSvnelovani geopolitikuri regionia.

yvela am winadadebas, maTi Sinaarsisa da maTi Semadgeneli sityvebis

raodenobis saxvadasvaobis miuxedavad, erTi ram aerTianebs: arc erTi

maTgani ar Seicavs iseT komponents, romelic Tavad SegviZlia ganvixiloT

rogorc damoukidebeli winadadeba. swored aseT winadadebebs vuwodebT

martivs. zogjer maT atomarul winadadebebsac uwodeben, xazi rom gausvan

im viTebas, rom maTgan SegviZlia avagoT sxva, ufro rTuli winadadebebi

iseve, rogorc molekula igeba atomisgan. am meore jgufs rTuli

winadadebi hqvia da maTi agebisaTvis Cveulebrivi, sametyvelo ena uamrav

saSualebas flobs, romlebic am enis sintaqsSia aRwerili. qvemoT, Cven am

saSualebaTa SezRuduli raodenobis ganxilviT SemovifarglebiT. es

bunebrivicaa, vinaidan Cven ara enis sintaqss vikvlevT, aramed garkveuli

sintaqsuri saSualebebiT agebuli winadadebebis urTierTdamokidebulebas.

amgvarad, fraza winadadebTa Soris arsebul damokidebulebebs, romelnic, mxolod da mxolod, am winadadebaTa agebis TaviseburebebiTaa gapirobebuli swored am azriT unda gavigoT. aqedan, sxvaTa Soris, erTi saintereso viTareba gamomdinareobs: vinaidan Cven gvainteresebs mxolod

winadadebaTa dakavSirebis Tavisebureba, anu is wesi, Tu rogor aris

winadadeba agebuli, amitom aranairi mniSvneloba ara aqvs im viTarebas, Tu

rTuli winadadebis komponentebi ra Sinaarsis matarebelia. Cveni interesis

sfero ara rTuli winadadebis Sinaarsi, aramed misi agebis sqemaa. Sedegad,

27 Cven yovelTvis avaridebT Tavs winadadebis Sinaars da mxolod im

saSualebebze mivmarTavT yuradRebas, riTac is igeba. am saSualebebs,

Cveulebriv, sawinadadebo makavSireblebs vuwodebT. es ukanasknelni, Tavis

mxriv, erTgvari operaciebia, romlebic, Tu ki maT winadadebebs mivuyenebT,

Sedegad isev winadadebas miviRebT. magaliTad, zemoTmoyvanili

winadadebebidan, iseTi makavSireblis gamoyenebiT, rogoricaa sityva ‘da’

SegviZlia sxvadasxva winadadebebi avagoT, miuxedavad imisa, rom Cveulebriv

sametyvelo enaSi zogierTi maTganis gamoTqmisagan, sul cota, Tavs

SevikavebdiT:

i. me vxedav mzes da TbilisSi aqtiuri politikuri cxovrebaa, ii. TbilisSi aqtiuri politikuri cxovrebaa da TbilisSi aqtiuri politikuri

cxovrebaa, iii. kavkasia mniSvnelovani geopolitikuri regionia da me vxedav mzes da sxv.

aqac mniSvnelovani ara maTi Sinaarsi, aramed maTi agebis Taviseburebaa, rac imaSi mJRavndeba, rom TiToeuli maTgani ‘da’ makavSireblis gamoyenebiT aris agebuli da am azriT, erTnairebia. Sinaarsisgan amdagvar damoukideblobas xazi rom gavusvan, propoziciur logikaSi e.w. sawinadadebo cvladebs iyeneben. es gaxlavT laTinuri anbanis asoebi, romlebic imdagvaradve gamoiyeneba, rogorc amas maTematikaSi akeTeben, im gansxvavebiT, rom ricxvebis nacvlad, cvladebis mniSvnelobad aq winadadebebi igulisxmeba. magaliTad, Tu winadadebas me vxedav mzes aRvniSnavT asoTi p, xolo winadadebas TbilisSi aqtiuri politikuri cxovrebaa

asoTi q, maSin winadadeba me vxedav mzes da TbilisSi aqtiuri politikuri

cxovrebaa aseT saxe miiRebs: ‘p da q’. bunebrivia, p-sa da q-s nacvlad nebismieri sxva TxrobiTi winadadeba SegviZlia vixmaroT da TiToeuls igive saxe eqneba. sinamdvileSi, TviTon gamosaxuleba ‘p da q’ aranair winadadebas ar warmoadgens, aramed winadadebad iqceva mxolod mas Semdeg, rac laTinuri asoebis nacvlad masSi qarTuli enis ronelime TxrobiTi winadadebaa (an ori sxvadasva winadadebaa) Canacvlebuli. qarTul enas imitom gavusviT xazi, rom ‘da’ makavSirebeli qarTul enas ekuTvnis. mis naclad rom gvqonoda Sinaarsobrivad igive, Tumca enobrivad gansxvavebuli makavSirebeli ‘and’, maSin inglisuri winadadebebi unda Cagvesva. es viTareba or rameze migvaniSnebs:

1. ‘p da q’ –s magvari gamosaxulebebi erTgavari yalibebi, formebia, romlebis winadadebad mxolod maTSi myofi cvladebis Sesabamisi enis winadadebebiT SenacvlebiT gardaiqmneba. maT xSirad, aseve, sawinadadebo formebad, an formulebad moixsenieben. es viTareba yovelTvis unda viqonioT mxedvelobaSi, miuxedavad imisa aSkarad vixmarT frazas sawinadadebo forma (an formulas) Tu ara.

2. imisaTvis, rom Cveni ganxilvebisas mxolod qarTul enaze ar viyoT mibmuli, rogorca Cans, makavSireblebisTvisac garkveuli simboloebia Semosatani, msgavsad maTematikisa, sadac. vTqvaT, ‘+’ aRniSnavs qarTulad Sekrebas, xolo inglisurad, add-s. Tumca gamosaxuleba a+b orive enaze erTi da igive Sinaarsis matarebelia.

28

amis mizezia, swored is rom a+b=b+a azriania, miuxedavad imisa Tu ra enaze wavikiTxavT mas.

ufro dawvrilebiT makavSireblebsa da maT Tvisebebze, aseve maTi saSualebiT agebul winadadebaTa urTierTdamokidebulebaze da mimarTebebze Semdeg paragrafebSi gveqneba saubari.

$2. winadadebaTa WeSmaritebiTi mniSvneloba. WeSmaritebiTi cxrilebi TxrobiT winadadebebs, garda imisa, rom isini SesaZloa martivi, an rTuli agebulebisa iyos, gaaCniaT kidev erTi Tavisebureba: maTi erTi nawili SeiZleba WeSmariti iyos, xolo meore nawili – mcdari. xSirad, winadadebaTa WeSmariteba an mcdaroba advili amosacnobia, zogjer ki amisaTvis fiqria saWiro. zogjer isec xdeba, rom gviWirs Tqma winadadeba WeSmaritia Tu mcdari ara imis gamo, rom Sesbamisi codna ar gagavaCnia, aramed imis gamo, rom winadadebaSi gamoTqmuli azri (debuleba) realobaSi arafers ar exeba, arafris mimarTaa gamoTqmuli. magaliTad, winadadeba kentavrs oTxi fexi aqvs. WeSmaritia is Tu mcdari? Tu miviCnevT, rom berZnuli miTologiis mixedviT, kentavri naxevrad cxenia, Tanac wels qvemoT, maSin TiTqo WeSmariti debuleba gamogviTqvams. magram Tu gavixsenebT, rom realobaSi aseTi arseba aravis Sexvedria da savaraudodac ar Sexvdeba, maSin gaugebaria, ras gaaCnia oTxi fexi? da Tu ki amis saxeli araferia, samarTliania vivaraudoT, rom aRniSnuli winadadeba mcdaria. sxvaTa Soris, msgavsi winadadebebis gaazrebas Rrma filososfiuri konotaciebi gaaCnia da sakmaod scildeba logikis mecnierebis farglebs. sxva dros rom amgvari gaugebrobani Tavidan aviciloT, ramodenime daSvebis Taobaze SevTanxmdeT:

a) winadadebani, romelTanac Cven saqme gveqneba, aucileblad an WeSmaritia, an mcdari

b) SeuZlebelia, romelime winadadeba erTdroulad mcdaric iyos da WeSmaritic

am daSvebebidan erTs gamoricxuli mesamis, xolo meores – winaaRmdegobis SeuZleblobis kanons uwodeben. msjeloba imaze, Tu am or kanons, romelic logikis ganviTarebis manZilze mniSvnelovan rols TamaSobda, ratom vuwodeT daSveba, sakmaod Sors wagviyvanda. gvinda ubralod xazi gavusvaT im garemoebas, rom logika sxva daSvebebidanac aigeba, oRond es iqneba ukve sxva logika da ara is, romlis aRwerasac vapirebdiT.

sxvaTa Soris, axla SegviZlia davsvaT sakmaod saintereso kiTxva: rogori iqneba rTuli winadadebis WeSmaritebiTi mniSvneloba, Tu ki cnobilia misi Semadgeneli winadadebebis WeSmaritebiTi mniSvnelobani? sxva sityvebiT, aris Tu ara raime kavSiri rTuli winadadebis WeSmariteba-mcdarobasa da misi komponentebis WeSmariteba-mcdarobas Soris? am kiTvaze rom pasuxi gavceT, kidev erT rameze SevTanxmdeT:

29

c) winadadebani, romelTanac Cven saqme gveqneba, mxolod da mxolod iseTi makavSireblebiTaa agebuli, rogoricaa ‘da’, ‘an’, ‘Tu – maSin’, ‘maSin da mxolod maSin’ , da ‘mcdaria, rom’. Sesabamisad, winadadebebs, romelic Cveni interersis obieqtia aqvT erT-erTi saxe Semdegidan: ‘. . . da ₋₋₋’, ‘. . . an ₋₋₋ ’, ‘Tu . . . , maSin ₋₋₋₋’, ‘. . . . maSin da mxolod maSin, Tu ₋₋₋₋’ da ‘mcdaria, rom ₋₋₋₋’, sadac wertilebiTa da tireebiT moniSnuli adgili raime WeSmarit an mcdar winadadebas gulisxmobs.

yvela am makavSirebels logikaSi Tavisi saxeli aqvs da maT specialuri simboloebiT aRniSnaven. aRniSnuli tipis winadadebebi yvelaze xSirad gvxdeba ara marto maTematikuri dasabuTebebis, aramed, saerTod, msjelobisas. amgvarad, es arCevani arc Tu mTlad nebismieria. ganvixiloT isini cal-calke.

$2.1 WeSmaritebiTi cxrilebi: koniunqcia. rTul winadadebas, romelic ‘da’ makavSireblis saSualebiTaa agebuli, Cveulebriv, koniunqcias uwodeben. sainteresoa, rom koniunqcia hqvia rogorc makavSirebels, aseve misi saSualebiT agebul winadadebasac. koniunqciuri winadadeba, SeiZleba iTqvas, erT-erTi yvelaze gavrcelebuli tipia winadadebebisa rogorc sametyvelo enaSi, aseve mecnieruli dasabuTebebisas. simbolo, romliTac

koniunqcias aRvniSnavT, aris: ∧ da is pirvelsa da meore winadadebs, an am winadadebaTa aRmniSvnel simboloebs Soris Tavsdeba. aRniSnul winadadebebsa da simboloebs koniunqciis wevrebs, anu koniunqtebs veZaxiT. magaliTad, zemoT motanil sawinadadebo formas ‘p da q’ –s amieridan ase CavwerT: (p∧q), sadac p-c da q-c koniunqciis wevrebs warmoadgens. Tumca literaturaSi sxva simboloebic ixmareba. TiToeuli simbolos arCeva avtoris gamovnebazea damokidebuli. raime principuli mosazreba am arCevans ar gansazRvravs. erTaderTi, rac avtors moeTxoveba, aris is, rom man simboloebi ar aurios da erTi da igive simbolo, erTi teqstis SigniT, erTi da igive mniSvnelobiT ixmaros. Cven mier SerCeuli simbolo erT-erTi yvelaze xSirad xmarebuli variantia. axla vcadoT da pasuxi gavceT Cvens kiTxvas koniunqciasTan mimarTebaSi: SegviZlia Tu ara koniunqciis WeSmaritebisa an mcdarobis dadagena mxolod misi Semadgeneli winadadebebis WeSmaritebisa an mcdarobis codnidan gamomdinare, ise rom amisaTvis ar gaxdes saWiro mTliani winadadebis Sinaarsis kvleva? pasuxs mivagnebT, Tu kargad davukvirdebiT Tavad koniunqciis raobas. ras gveubneba koniunqcia? koniunqcia gveubneba, rom pirveli winadadebiT gamoTqmuli mosazreba da meore winadadebiT gamoTqmuli mosazreba erTdroulad WeSmariti mosazrebebi gaxlavT. sxva sityvebiT, me Tu

gamovTqvam (p∧q), me miviCnev da vcdilob dagarwmunoT, rom adgili aqvs rogorc p-Ti aRniSnul viTarebas, aseve im viTarebas, romelic aRniSnulia q-Ti. im SemTxvevaSi, Tu me ganzrax ar vityuebi, anda ar vcdebi uneblied, es asecaa. magram ras niSnavs adgili aqvs rogorc p-Ti, ise q-Ti aRniSnul viTarebas? es niSnavs mxolod imas, rom WeSmaritia rogorc p, aseve q. amgvarad, SegviZlia davaskvnaT, rom rodesac viciT, rom mocemuli koniunqciis orive koniunqti WeSmaritia, koniunqcia WeSmaritia, rasac ar

30 unda Seexebodes misi Sinaarsi. axla davuSvaT, rom erT-erTi koniunqti mocemul koniunqciaSi mcdaria da Cven am mcdarobis dadgena movaxerxeT. bunebrivia, vifiqroT, rom mTlianad koniunqciac mcdar debulebas warmogvidgens. es asecaa. rodesac me vcdilob dagarwmunoT imaSi, rom

adgili aqvs (p∧q)-s, xolo aRmoCndeba, rom erT-erTi ar asaxavs imas, rac

xdeba, Tqven sruli ufleba gaqvT miTxraT, rom (p∧q) (mTlianobaSi)

mcdaria. aseve bunebrivia, (p∧q) mcdarad miviCnioT, rodesac mcdaria p da mcdaria q.

sainteresoa, rom yvelafer amas cxrilis saxe SeiZleba mieces. cifrebiT 1 da 0 aRvniSnoT Sesabamisad ‘WeSmariti’ da ‘mcdari’. maSin, p–ze SegviZlia vTqvaT, rom is SeiZleba iRebdes mniSvnelobas 1, an iRebdes mniSvnelobas 0. magram igives Tqma SegviZlia q-zec. amgvarad, variantebi, romelic

SesaZlebelia rom (p∧q)-sTvis gavvixiloT, gamogviva oTxi. keZod

i. (1 ∧ 1) ii. (1 ∧ 0)

iii. (0 ∧ 1) iv. (0 ∧ 0)

sxva sityvebiT,

WeSmaritia p ∧ WeSmaritia q (i)

WeSmaritia p ∧ mcdaria p (ii)

mcdaria p ∧ WeSmaritia p (iii)

mcdaria p ∧ mcdaria q (iv). axla, vinaidan viciT, Tu TiToeuli kombinacia ras gavZlevs, advilad avagebT cxrilsac, romelsac koniunqciis WeSmaritebiT cxrils vuwodebT. cxrilis agebisas vixelmZRvaneloT SemdegiT: jer davadgenT WeSmaritebiT mniSvnelobebs koniunqtebisaTvis, xolo Semdeg, mTeli koniunqciisaTvis. Sedegad, gveqneba sami oTx-oTx striqoniani paraleluri sveti. TiToeuli satriqonis pirveli ori cifri (1 an 0) gvaZlevs koniunqtebis WeSmaritebiT mniSvnelobas, xolo bolo cifri, konunqtTa mocemuli mniSvnelobisaTvis, Tavad koniunqciis mniSvnelobas iZleva. cxrili 1. koniunqcia

p q p∧q 1 1 1 1 0 0 0 1 0 0 0 0

31 es cxrili sainteresoa imiTac, rom, praqtikulad, iZleva saSualebas yovelTvis vicodeT koniunqciis WeSmariteba, Tu ki misi komponentebis WeSmaritebiTi fasi gvaqvs dadgenili. metic, aseTi codnis misaRebad sxva damatebiTi codna ar aris saWiro. aseve sainteresoa is garemoebac, rom koniunqcia WeSmaritia maSin daAmxolod maSin, Tu ki misi orive koniunqti WeSmaritia. sxva piroba koniunqciis WeSmaritebas ar adevs.

$2.2. WeSmaritebiTi cxrilebi: disiunqcia. axla SevxedoT, ras gvaZlevs makavSirebeli ‘an’. rTul winadadebas, romelic ‘an’ makavSireblis saSualebiTaa agebuli, disiunqcias uwodeben, xolo mis wevrebs –

disiunqtebs. simbolo ‘∨’ , Cveulebriv, swored disiunqciis aRsaniSnavad ixmareba da, koniunqciis msgavsad, isic disiunqtebs Soris Tavsdeba. maSasadame, gamosaxulebas, romelsac disiunqcias veZaxiT, eqneba Semdegi

saxe: (p∨q). SegviZlia Tu ara disiunqciisTvisac avagoT iseTive cxrili, rogorc es koniunqciis SemTxvevaSi gamogvivida. didi albaTobiT, rogorc Cans, ki. Tumca, amave albaTobiT, savaraudod, es cxrili gansxvavebuli unda iyos. sxvaobas koniunqciasa da dasiunqcias Soris maTi enaSi gamoyenebis semantikuri Tavisebureba ganapirobebs. raze migvaniSnebs ‘an’ makavSireblis saSualebiT dakvSirebuli ori winadadeba? imaze xom ara, rom ori xdomilebidan, romlebic am winadadebebiTaa aRwerili, erT-erTi mainc unda realizdes? magaliTad, rogor SevxedoT winadadebas

TbilisSi aqtiuri politikuri cxovrebaa an kavkasia mniSvnelovani geopolitikuri regionia ?

rodis iqneboda is WeSmariti? cxadia, Tu TbilisSi aranairi poltikuri aqtioba ar mimdinareobs, da arc kavkasia warmoadgens msoflios wamyvan saxelmwifoTa politikuri interesis sferos, maSin mTliani winadadeba mcdaria. sxva sityvebiT, im SemTxvevaSi, rodesac orive disiunqti mcdaria, disiunqcia aucileblad mcdari iqneba.

Cveulebriv, aseve ar badebs SekiTxvebs viTareba, rodesac TbilisSi aqtiuri politikuri cxovreba realobaa, xolo kavkasia mainc ar warmoadgens msoflios wamyvan saxelmwifoTa politikuri interesis sferos. amgvar viTarebas Cven imis dasturad miviCnevdiT, rom mTlianobaSi, zemoT motanili gamonaTqvami WeSmariti winadadebaa.

aseve eWvs ar gamoiwvevda situacia, rodesac TbilisSi politikuri cxovreba minelebulia, xolo kavkasia mainc warmoadgens msoflios wamyvan saxelmwifoTa politikuri interesis sferos, vTqvaT, Tundac Suaaziuri energoresursebis tranzitis gaadvilebis TvalsazrisiT. vinaidan, am SemTxvevaSic ,miviCnevT, rom mTlianobaSi, zemoT motanili gamonaTqvami WeSmariti winadadebaa. sxva sityvebiT, disiunqcia WeSmaritia, Tu ki misi erT-erTi disiunqti aris WeSmariti. magram ra xdeba im SemTxvevaSi, Tu ki WeSmaritia orive disiunqti? sxva sityvebiT, ‘an’ makavSireblis xmareba aucileblobiT xom ar gamoricxavs erT-erTi xdomilebis realizaciis SesaZleblobas, Tu ki

32 viciT, rom meore ukve realizebulia? Cveni magaliTis mixedviT Tu vimsjelebT, TiTqos, es ase ar unda iyos: marTlac, kavkasiisadmi geopolitikuri interesi ar aris damokidebuli TbilisSi politikuri amindis cvalebadobaze. Tumca, aseve SesaZlebelia iseTi magaliTebis motanac, rodesac erTi komponentis WeSmariteba sworedac, rom meore komponentis mcdarobas gulisxmobs. magaliTad, nebismieri naturaluri

ricxvebisaTvis x da y, Tu ki isini sxvadasxava ricxvebia, (x>y) ∨ (y>x) aris WeSmariti winadadeba. magram, (x>y) –is WeSmariteba, cxadia, gamoricxavs (y>x) –is WeSmaritebas. rogor moviqceT, rom arc erTi es Sinaarsi ar davkargoT? enobriv doneze SegviZlia SevTanxmdeT, rom ‘an’ makavSireblis xmareba SevzRudoT mxolod iseTi SemTxvevebiT, sadac iseTi mkacri piroba, rom mxolod erTi disiunqti unda iyos WeSmariti, ar moqmedebs. es, ase vTqvaT, aragamomricxavi disiunqciaa. xolo im SemTxvvebisTvis, rodesac piroba aseTi mkacria, makavSireblad vixmaroT ‘an - an’ (gamomricxavi

disiunqcia). Sesabamisad, simbolo ‘∨’, da saxeli disiunqcia SevinarCunoT pirveli SemTxvevisaTvis, anu aragamomricxavi disiunqciisaTvis. meore SemTxvevisTvis ki gamoxatvis sxva saSualebebi vipovoT. ra Tqma unda, es ar aris erTaderTi gamosavali, Tumca sakiTxisadmi midgomis yvelaze gavrcelebuli da moxerxebuli formaa. aRniSnulidan gamomdinare, advilad SegviZlia avagoT WeSmaritebiTi cxrili disiunqciisaTvis. amisaTvis Cven imave meTods gamoviyenebT, rac koniunqciis cxrilis asagebad vixmareT.

cxrili 2. disiunqcia

p q p∨q 1 1 1 1 0 1 0 1 1 0 0 0

am cxrilidan ara marto kargad Cans, Tu rogor vuyurebT disiunqcias, aramed isic, Tu ra ganasxvavebs mas koniunqciisganac: disiunqcia mcdaria maSin da mxolod maSin, rodesac misi orive disiunqti mcdaria. sxva piroba disiunqciis mcdarobas ar adevs.

$2.3. WeSmaritebiTi cxrilebi: uaryofa. uaryofas vuwodebT iseT frazebs, rogoricaa ‘mcdaria, rom’, an ‘ar aris marTali, rom’, an ‘tyuilia, rom’ da sxv. aRniSnuli frazebi, romelic raime mocemul winadadebas winidan daerTvis, mianiSnebs imaze, rom am winadadebiT gamoTqmuli mosazreba mcdaria, ar aris WeSmariti. mkacrad Tu vimsjelebT, uaryofa makavSirebels ar warmoadgens, vinaidan misi saSualebiT ar xdeba ori, an meti winadadebis dakavSireba, aramed erTi, konkretulad mocemuli Sinaarsis uaryofa. miauxedavad amisa, imis gamo, rom uaryofis gamoyenebis Sedegad isev mcdar an WeSmarit winadadebas viRebT, mas makavSireblebTan erTad ganvixilavT. xSirad, aseTi midgoma mosaxexebelia. iqidan

33 gamomdinare, rac vTqviT, advili misaxvedria, rom WeSmariti winadadebis uaryofa mcdaria, xolo mcdari winadadebis uaryofa – WeSmariti. uaryofasac, iseve rogorc sxva logikur makavSirebels, gansakuTrebuli

simboloTi aRniSnaven. Cven amisaTvis vixmarT simbolos ‘¬’, romelsac uarsayofi winadadebis, anda misi gamomsaxveli simbolos win davwerT.

amrigad, Canaweri ‘¬p’ aRniSnavs, rom ‘mcdaria p’. uaryofisTvis WeSmaritebiTi cxrilis asagebad aRar dagvWirdeba oTxi saxavdasxva SemTxvevis ganxilva, vinaidan saqme oris nacvlad, erT uarsayof winadadebasTan gvaqvs.

cxrili 3. uaryofa

p ¬p 1 0 0 1

amgvarad, winadadeba, romelic uaryofis simbolos Seicavs, WeSmaritia maSin da mxolod maSin, rodesac uarsayofi winadadeba aris mcdari. es viTareba TanxmobaSia Cvens intuiciasTan da, sxvaTa Soris, eyrdnoba Cveni daSvebebidan erT-erTs, kerZod, (a)-s (ix. §2). $2.4. WeSmaritebiTi cxrilebi: implikacia. implikacias iseT rTul winadadebas vuwodebT, romelsac, zogadad, Semdegi forma gaaCnia: ‘Tu . . . , maSin ₋₋₋₋’. mas zogjer pirobiT winadadebas, anda gamomdinareobasac uwodeben. Tumca, am bolo sityvis xmarebas garkveuli sifrTxile Wirdeba, vinaidan is SeiZleba mizez-Sedegobriv kavSirsac gulisxmobdes, rac implikaciis ganxilvis SemTxvevaSi, Cven, sazogadod, gvinda, rom Tavidan aviciloT. ‘Tu’-s Semdeg myof winadadebas antecedenti hqvia, xolo winadadebas, romelic sityva ‘maSin’-is Semdeg gvxvdeba hqvia konseqventi. pirveli terminis sinonimebia piroba, wanamZRvari, daSveba, varaudi. meore terminis sinonimebad xmaroben Sedegi, daskvna. implikacia, iseve rogorc aqamde ganxiluli rTuli winadadebebi, xSirad ixmareba rogorc yoveldRiur metyvelebaSi, aseve mecnierebaSi. gansakuTrebiT bevri magaliTi implikaciuri winadadebebisa SegviZlia maTematikaSi moviZioT. praqtikulad, maTematikur TeoremaTa didi nawili, aSkarad an farulad, implikacias warmoadgens. simbolo, romelic implikaciis aRsaniSnavad metwilad ixmareba, aris ‘⊃’. masac, iseve rogorc koniunqciisa da disiunqciis SemTxvevaSi, implikaciis Semadgenel winadadebaTa Soris vaTavsebT. amgvarad, gamosaxuleba ‘p⊃q’ ikiTxeba rogorc ‘Tu p, maSin q’. axla vnaxoT, Tu rogor mivudgeT WeSmaritebiTi cxrilis agebas implikaciisTvis? intuiciurad, implikaciis gamoTqmisas, Cven vcdilobT vaCvenoT, rom piroba gulisxmobs Sedegs. sxva sityvebiT, antecedentis WeSmariteba gansazRvravs konseqventis WeSmaritebasac. xolo rodesac konseqventi mcdaria, es imis niSani unda iyos, rom antecedentic mcdari yofila. yovel SemTxvevaSi, Cven ar gvinda, rom WeSmariti pirobidan

34 mcdari Sedegi gvqondes. es garemoeba gvkarnaxobs, rom ZalaSi unda iyos Semdegi sami piroba:

i. Tu antecedentic da konseqventic WeSmaritia, implikaciac SeSmaritia. ii. Tu antecedenti WeSmaritia, xolo konseqventi mcdari, maSin

implikaciac mcdari iqneba iii. Tu mcdaria rogorc antecedenti, aseve konseqventi, implikacia

WeSmaritia.

magram ra viTarebaa, rodesac antecedenti mcdaria, xolo konseqventi WeSmariti? samwuxarod, aqac imave fsiqologiur bariers vawydebiT, rogorsac disiunqciis ganxilvisas gadavawydiT. Cveni Cveva, vixmaroT ena rogorc komunikaciis ZiriTadi saSualeba, ganapirobebs enobriv gamosaxulebaTa maqsimalur Sinaarsobriv datvirTulobas. imlikaciis SemTxvevaSi es pirobasa da Sedegs Soris nagulismeb erTgvar kavSirSi mJRavndeba. zogjer es mizezobrivi, zogjer ki Sinaarsobrivi kavSirebia. yovel SemTxvevaSi, Cven ar varT ganwyobili, rom im SemTxvevebSi, rodesac aseTi kavSiri cxadad ara Cans, implkaciuri winadadeba WeSmaritad miviCnioT. aseve naklebad varT ganwyobili WeSmaritad miviCnioT implikacia, rodesac dadasturebulad viciT, rom antecedenti mcdaria, xolo konseqventi WeSmariti. nebismieri argumentacia am fsiqologiur bariers gadaawydeba da damajerblobis xarisxs dakargavs. amitom gza aqac erTia: unda vipovoT iseTi varianti, romelic yvelaze axloa imasTan, rac gvinda da Semdeg, am variantze SevTanxmdeT, ramdenadac ar unda gvexamuSebodes misi miReba. Tu gavixsenebT, erTis mxriv, rom winadadebaTa Sinaarsobriv ganxilvaze uari gvaqvs naTqvami, xolo, meores mxriv, imas, rom ar gvinda, WeSmariti pirobidan mcdari Sedegi gvqondes, xolo danarCens gadamwyveti mniSvneloba ara qvs, advilad SevTanxmdebiT imazec, rom rodesac antecedenti mcdaria, xolo konseqventi WeSmariti, implikacia WeSmaritia. amgvarad, implikaciisaTvis gveqneba Semdegi cxrili:

cxrili 4. implikacia

p q p⊃q 1 1 1 1 0 0 0 1 1 0 0 1

amgvarad, winadadeba, romelic implikaciis simbolos Seicavs, mcdaria mxolod maSin, rodesac antecedenti WeSmaritia, xolo konseqventi - mcdari.

$2.5. WeSmaritebiTi cxrilebi: eqvivalentoba. rTul winadadebas, romelic misi Semadgeneli winadadebebis igiveobriobas daadgens, vuwodebT eqvivalentobas. eqvivalentobis sinonimebia tolfasoba, ormxrivi gamomdinareoba da a.S. damakavSiremeli fraza, romelic Cveulebriv

35 moixmareba, gansakuTrebiT maTematikaSi, aris ‘maSin da mxolod maSin’. Tumca arsebobs misi iseTi Semcvlelebi, rogorisaa, magaliTad, ‘sakmarisi da aucilebeli’, an misi romelime sinonimi. simbolo, romelic amisaTvis ixmareba, Semdegia: ‘≡’ da is, iseve rogorc sxva SemTxvevebSi, rodesac saqme ori winadadebis dakavSirebasTan gvqonda, winadadebebs Soris iwereba. ase, rom gamosaxuleba (p≡q) ubralod niSnavs imas, rom p da q igiveobriv winadadebebadaa miCneuli. Tumca, igiveobas sxvadasxva ganzomileba aqvs. winadadebebi SesaZlebelia igiveobrivni iyvnenen im azriT, rom erTsa da imave Sinaarss gamoTqvamdnen, oRond enis sxvadasxva gamomsaxvelobiTi saSualebebis gamoyenebiT. sxvaTa Soris, es erT-erTi motivaciaa imisaTvis, rom iveobis gamomTqmel winadadebebs yuradReba mivaqcioT. aseve SesaZlebelia igiveoba exebodes ara Sinaars, aramed im WeSmaritebiT mniSvnelobas, romelic p-Ti da q-Tia gamoTqmuli. arsebobs kidev sxva ganzomilebani, magram Cven aq SevCerdeT, vinaidan CvenTvis saintereso swored rTuli winadadebis WeSmariteba-mcdarobis damokidebulebaa Semadgeneli winadadebebis WeSmariteba-mcdarobasTan. Tu am viTarebidan gamovalT, advilad davrwmundebiT, rom (p≡q) WeSmariti SeiZleba iyos mxolod maSin, rodesac p-sac da q-sac erTnairi WeSmaritebiTi maxasiaTeblebi aqvs, kerZod, orive erTdroulad an WeSmaritia, an mcdari. sxva SemTxvevaSi eqvivalentobis gamomTqmeli winadadeba iqneba mcdari. amaze dayrdnobiT ar unda gagviWirdes Sesabamisi cxrilis agebac.

cxrili 5. eqvivalentoba

p q p≡q 1 1 1 1 0 0 0 1 0 0 0 1

$3. damokidebulebani sawinadadebo makavSireblebs Soris.

mas Semdeg, rac gavecaniT ZiriTad makavSirebelTa rols rTuli

winadadebebis agebaSi da gavarkvieT maTi Taviseburebani WeSmariteba-

mcdarobis TvalsazrisiT, bunebrivia daisvas kiTxva, arsebobs Tu ara maT

Sorisac raime saxis damokidebuleba? sxva sityvebiT, rogor

urTierTmoqmedeben isini, Tu ki aseTi ram saerTod SesaZlebelia? Aam

kiTxvis kanonierebaze, sul cota ori ram migvaniSnebs. jer erTi, analogia

maTematikasTan: viciT, rom iseTi matematikuri mudmivebi, rogoricaa +, -, ×

da : SegviZlia ara marto erTmaneTis meSveobiT gamovxatoT, aramed erTi

da imave meTematikuri frazis (formulis) SigniT, maTi saSualebiT

garkveuli manipulaciebi gavakeToT, rac sabolood, gamoTvlebs gviiolebs.

magaliTad, viciT rogor moviqceT, rodesac jams raime ricxvze

vamravlebT, vinaidan viciT, rom +-s da ×-s gaaCnia Semdegi Tavisebureba:

36 c ×(a+b) = (c×a) + (c×b). raime msgavss xom ar SeiZleba qondes adgili

sawinadadebo mkavSireblebis SemTxvevaSi? am azrs isic aZlierebs, rom

‘p⊃q’ msagavs da sxva formulebSic, rogorc p, ise q SesaZlebelia, rom

nebismieri agebulebisa iyos. anu, viTareba, rodesac erTi formulis SigniT

sxvadasva makavSireblebiT agebuli nawilebi gvxdeba, savsebiT bunebrivi da

misaRebi Cans. Tumca, sanam sakiTxs gavarkvevdeT, nebismieri formulis

ganxilvisas, Cven garantirebuli unda viyoT im azriT, rom aRniSnuli

formula yovelTvis gamoxatavs raime dasrulebul winadadebas, Tu ki

masSi laTinuri anbanis asoebis nacvlad, anu sawinadadebo cvladebis

nacvlad, romelime enis (vTqvaT, igive qarTulis) winadadebebs

CavanacvlebT. sxvanairad rom vTqvaT, Cven garantirebuli unda viyoT, rom

Cveni formulebi sworad aris agebuli. Aam garantias erTaderTi ram

uzrunvelyofs: Cven unda SevTanxmdeT wesebze, romliTac formulebs

vadgenT sawinadadebo makavSireblebis gamoyenebiT da mxolod aseTnairad

agebuli formulebi miviCnioT formulebad. isev maTematikas rom

mivmarToT analogiisTvis, raRac msgavss iqac vnaxavT. magaliTad, aravin

CaTvlida azrianad c ab) = + c×a)c×b) gamosaxulibis ganxilvas, anda saerTod

ar miiCnevda mas formulad, maSin rodesac imave niSnebis saSualebebiT

agebuli gamosaxuleba c ×(a+b) = (c×a) + (c×b) ariTmetikis erT-erT ZiriTad debulebas gamoxatavs. amgvarad, SevTanxmdeT Semdegze:

3.1. winadadebaTa (formulaTa) asagebad gvaqvs Semdegi saSualebebi:

• laTinuri anbanis patara asoebi, indeqsebiT(an maT gareSe), romlebsac

martiv an rTul TxrobiT winadadebaTa aRsaniSavad vixmarT.

• sawinadadebo makavSireblebi ¬, ∧, ∨, ⊃, ≡ • marjvena da marcxena mrgvali an kvadratuli frCxilebi (, ), [, ]

romlebic sasveni niSnebis rols asruleben.

• garda zemoT CamoTvlilisa, winadadebaTa (formulaTa) asagebad sxvas

arafers ar vixmarT.

3.2. Canaweri, romelic (3.1)-Si miTiTebuli simboloebiTaa Sedgenili,

warmoadgens formulas mxolod im SemTxvevaSi, Tu ki is akmayofilebs

Semdegi pirobebidan erT-erTs:

• is aris calke mdgomi laTinuri anbanis erT-erTi aso, indeqsiT, an

mis gareSe, romelic (3.1)-is pirobaSia nagulisxmebi.

• Tu raime p da q aris formula, maSin aseve formulebia ¬(p), ( p ∧q),

(p ∨q), (p⊃q), (p ≡q) .

• arc erTi sxva gamosaxuleba, romelic zemoT gansazRvrul pirobebSi

ar jdeba, anu, sxva ram pirobebze dayrnobiT aris agebuli, formulas

ar warmoadgens.

37

axla SegviZlia Cvens ZiriTad Temas davubrundeT da vnaxoT, Tu ra

damokidebulebani SeiZleba aRmoCndes Tavad makavSireblebs Soris.

daviwyoT, erTi SexedviT, martivi SemTxveviT, uaryofiT. ras gvaZlevs am

kuTxiT ¬(p)? Tu p martivia, pasuxi wina paragrafSi mocemul cxrilSi

(cxrili 3) unda veZioT. magram rasTan gvaqvs saqme, rodesac p rTuli

winadadebaa? Cveni SeTanxmebis Tanaxmad, Tu ki is rTulia, mas SeiZleba

hqondes Semdegi xuTi saxidan erT-erTi

i. p TviTon warmoadgens uaryofas

ii. p warmoadgens koniunqcias

iii. p warmoadgens disiunqcias

iv. p warmoadgens implikacias

v. p warmoadgens eqvivalentobas

ganvixiloT es SemTxvevebi:

i. p TviTon warmoadgens uaryofas. maSin ¬(p) Semdeg saxes miiRebs, ¬(¬r) sadc ¬r aris p. vnaxoT, am SemTxvevaSi, ras mogvcemda Cveni cxrili,

romelic uaryofisaTvis SevadgineT

r ¬r ¬¬r 1 0 1 0 1 0

Zalian saintereso rames mivageniT: Tu winadadebas orjer vuaryofT, isev

imave winadadebas miviRebT. Ees metad mniSvnelovani kanonia da mas ormagi

uaryofis kanons eZaxian. simbolurad is Semdegnairad SegviZlia CavweroT:

¬¬p≡ p.

ii. p warmoadgens koniunqcias. . maSin ¬(p) Semdeg saxes miiRebs, ¬(r∧s) sadc (r∧s) aris p. davkvirdeT am winadadebis Sinaarss. is gveubmeba,

rom koniunqcia (r∧s) aris mcdari. Ees ki, Tavis mxriv, Tu koniunqciisTvis agebul cxrils gavixsenebT, unda niSnavdes imas, rom

mcdaria misi an erTi koniunqti, an meore koniunqti, an orive erTad.

rac Semdegnairad Caiwereba (¬r∨¬s). sxva sityvebiT, koniunqciis uaryofa TiToeuli koniunqtis uaryofaTa disiunqcia yofila. esec

mniSvnelovani debulebaa da is de morganis kanonis saxeliTaa

cnobili. gamoviyenoT cxrili da davrwmundeT mis sisworeSi. Aanu

vnaxoT, rom, rogoric ar unda iyos r da s, ¬(r∧s)≡ (¬r∨¬s) aris WeSmariti debuleba.

38

r s ¬r ¬s (r∧s) ¬(r∧s) (¬r∨¬s) 1 1 0 0 1 0 0 1 0 0 1 0 1 1 0 1 1 0 0 1 1 0 0 1 1 0 1 1

axla, Tu am cxrilSi bolo or svets SevadarebT, davrwmundebiT

Cveni debulebis sisworeSi.

iii. axla, vTqvaT, p aris (r∨s). bunebrivia, wina SemTxvevis ganxilvis

Semdeg, gagviCndes imnairive Sedegis molodini. sxva sityvebiT, ¬(r∨s)≡ (¬r∧¬s) . mivmarToT cxrils.

r s ¬r ¬s (r∨s) ¬(r∨s) (¬r∧¬s) 1 1 0 0 1 0 0 1 0 0 1 1 0 0 0 1 1 0 1 0 0 0 0 1 1 0 1 1

aqac, bolo ori svetis Sedareba, gvarwmunebs Cveni debulebis

sisworeSi. sxvaTa Soris, esec de morganis kanonia.

iv. vnaxoT, ra situacia gveqneba, Tu ki implikacias vuaryofT. sxva

sityvebiT, ra unda iyos ¬(p⊃q)? Tu implikaciis daxasiaTebas

gavixsenebT, mivxvdebiT, rom misi uaryofa ubralod unda niSnavdes

misi antecedentis WeSmaritebasa da konseqventis mcdarobas. anu,

ZalaSi unda iyos Semdegi formula ¬(p⊃q)≡(p∧¬q). SevamowmoT cxrilis meSveobiT.

p q ¬q (p⊃q) ¬(p⊃q) (p∧¬q) 1 1 0 1 0 0 1 0 1 0 1 1 0 1 0 1 0 0 0 0 1 1 0 0

aRniSnuli garemoeba imiTac aris saintereso, rom Tu de morganisa da

ormagi uaryofis kanonebs moviSveliebT, advilad davrwmundebiT, rom

(p⊃q)≡(¬p∨q), rasac, Tavis mxriv, ufro saintereso Sedegamde mivyavarT:

39 rogorca Cans, Cven Tavisuflad SegveZlo implikaciis simbolos gareSe

masSi nagulisxmebi Sinaarsi gamogvexata. v. da bolos, vnaxoT ras warmoadgens eqvivalentobis uaryofa. cxrilis

gareSec, advili misxvedria, rom aseTi winadadeba mxolod maSin

iqneba WeSmariti, rodesac mis komponentebs sxvadasxva WeSmaritebiTi

mniSvnelobani gaaCnia. magram es xom maSin xdeba, rodesac erTi

komponenetis WeSmariteba gamoricxavs meore komponentis WeSmaritebas

da piriqiT. es ki zustad is aris, rasac makavSirebeli ‘an – an’

gulisxmobda. Aase, rom eqvivalentobis uaryofa gamomricxavi an-is

tolfasi yofila.

amavenairad, sxvaTa Soris, SegviZlia yvela makavSirebeli gavsinjoT, rac

saSualebas mogvcems bevri saintereso kavSiri davinaxoT maT Soris. Tumca

amjerad, amas Tavs avaridebT, vinaidan aRniSnuli, ZiriTadad, meqanikur

samuSaos Seadgens, da savarjiSos saxiT SeiZleba gakeTdes.

axla yuradReba mivaqcioT Semdeg viTarebas: zemoT motanili cxrilebis

bolo ori svetis Sedarebisas, orive warmodgenili formula erTsa da

imave WeSmaritebiT mniSvnelobas iRebda miuxedavad imisa Tu rogori

mniSvnelobebi miewereboda maT atomarul Semadgenlebs, ramac mogvca,

sxvaTa Soris, saSualeba dagvewera, rom ¬¬p≡ p, ¬(r∧s)≡ (¬r∨¬s) , ¬(r∨s)≡ (¬r∧¬s) da ¬(p⊃q)≡(p∧¬q). magram Tu yuradrebas TviT am eqvivalentobebze

gadavitanT, erT saintereso moments aRmovaCenT. isini mudmivad WeSmariti

formebia im azriT, rom rogori mniSvnelobanic ar unda mivusadagoT maT

atomarul Semadgenlebs, isini mainc WeSmaritebi iqnebian. amaSi rom

davrwmundeT, zemoTmotanil cxrilebs davamatoT kidev TiTo sveti da

Sedegs SevxedoT.

¬¬p≡ p

p ¬p ¬¬p ¬¬p≡ p 1 0 1 1 0 1 0 1

¬(r∧s)≡ (¬r∨¬s)

r s ¬r ¬s (r∧s) ¬(r∧s) (¬r∨¬s) ¬(r∧s) ≡ (¬r∨¬s) 1 1 0 0 1 0 0 1 1 0 0 1 0 1 1 1 0 1 1 0 0 1 1 1 0 0 1 1 0 1 1 1

40

¬(r∨s)≡ (¬r∧¬s)

r s ¬r ¬s (r∨s) ¬(r∨s) (¬r∧¬s) ¬(r∨s) ≡ (¬r∧¬s) 1 1 0 0 1 0 0 1 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 0 1 1 0 1 1 1

¬(p⊃q)≡(p∧¬q)

p q ¬q (p⊃q) ¬(p⊃q) (p∧¬q) ¬(p⊃q) ≡ (p∧¬q) 1 1 0 1 0 0 1 1 0 1 0 1 1 1 0 1 0 1 0 0 1 0 0 1 1 0 0 1

amdagvar sawinadadebo formebs, romlebic maTi atomaruli Semadgenlebis

WeSmaritebiTi mniSvnelobis miuxedavad, yovelTvis WeSmaritia, logikaSi

tavtologias eZaxian. tavtologiaze SeiZleba iTqvas, rom is WeSmaritia

Tavisi formis gamo. sxva sityvebiT, is iseTnairadaa agebuli, rom ra

Sinaarsic ar unda mivaniWoT mis komponentebs, is mainc WeSmariti iqneba.

tavtologiis uaryofas uwodeben winaaRmdegobas. bunebrivia, winaaRmdegoba

yovelTvis mcdaria.

$4. Ppropoziciuri logikis ZiriTadi kanonebi.

qvemoT CamovTvliT im ZiriTad damokidebulebebs, romelTac, xSirad,

propoziciuri logikis kanonebs eZaxian.zogierTi maTgani ukve nacnobia

CvenTvis, xolo danarCenebis Sinaarsi advili gasaazrebelia. Yyovel

SeTxvevaSi, maTi tavtologiurobis Semowmeba advilad SeiZleba

WeSmaritebiTi cxrilebiT.

((p⊃q)∧p)⊃q

Mmodus ponensi

((p⊃q)∧¬q)⊃¬p

Mmodus tolensi

(p⊃q)⊃((q⊃r)⊃(p⊃r))

hipoTezuri silogizmi

41 ((p∨q)∧¬p)⊃q

disiunqciuri silogizmi

((p⊃q)∧(r⊃s))⊃((p∨r)⊃(q∨s))

konstruqciuli dilema

((p⊃q)∧(r⊃s))⊃((¬q∨¬s)⊃(¬p∨¬r))

destruqciuli dilema

(p∧q)⊃p (p∧q)⊃q

simplifikacia (gamartiveba)

p⊃(p∨q) q⊃(p∨q)

Ddisiunqtis damateba

(p⊃q)⊃((p⊃r)⊃(p⊃(q∧r)))

kompozicia

(¬¬p) ≡ p

ormagi uaryofis kanoni

(p∧q) ≡ (q∧p) gadanacvlebis (komutaciis) kanoni koniunqciisaTvis

(p∨q) ≡ (q∨p) gadanacvlebis kanoni disiunqciisaTvis

(p≡q) ≡ (q≡p) gadanacvlebis kanoni eqvivalentobisTvis

((p∧q)∧r)≡(p∧(q∧r)) dajgugebis (asociaciurobis) kanoni koniunqciisaTvis

((p∨q)∨r) ≡(p∨(q∨r))

dajgugebis kanoni disiunqciisaTvis

(p∧(q∨r)) ≡ ((p∧q)∨(p∧r)) ganawilebis (distribuciis) kanoni koniunqciisa disiunqciis mimarT

(p∨(q∧r)) ≡ ((p∨q)∧(p∨r)) ganawilebis kanoni disiunqciisa koniunqciis mimarT

¬(p∧q)≡(¬p ∨¬q) ¬(p∨q) ≡ (¬p ∧ ¬q)

de morganis kanonebi

(p⊃q) ≡ (¬p v q) ¬ (p⊃q) ≡ (p ∧ ¬q)

implikaciis eliminaciis (mocilebis) kanonebi

(p⊃q) ≡ (¬q ⊃ ¬p) (q⊃p) ≡ (¬p⊃ ¬q)

Kkontrapoziciis kanonebi

42 (p⊃q) ≡ ((p ⊃q) ∧ ( q ⊃ p)) ekvivalentobis eliminaciis kanoni

p ∨ ¬p gamoricxuli mesamis kanoni

¬(p∧ ¬p) winaaRmdegobis SeuZleblobis kanoni

kargi iqneba, Tu am formulebis SinaarsSi gaverkveviT, da im principebs

davimaxsovrebT, rac maTi saSualebiTaa gamoTqmuli. am principebze

dayrdnobiT bevri debulebis dasabuTeba martivdeba.

43 Tavi III. predikatTa logikis sawyisi cnebebi da ZiriTadi

kanonebi

$1. ras vuwodebT predikatTa logikas?

mas Semdeg, rac gavecaniT propoziciuri logikis ZiriTad cnebebsa da

gamomsaxvelobiT saSualebebs, TiTqos advilad unda SevZloT Cveulebrivi

sametyvelo eniT gamoTqmuli argumentebis simboloebis enaze

“gadaTargmna”, ise, rom Tu ki Cveni argumentacia swori iyo, simbolur

enaze Targmnili, is tavtologiad gardaiqmnas. zogjer es marTlac asea.

ganvixiloT magaliTi:

Tu Tovs, maSin ar wvims

wvims

maSasadame ar Tovs.

propoziciuri logikis gamomsaxvelobiT saSualebebs Tu moviSveliebT,

vnaxavT, rom aq gvaqvs ori sabazo winadadeba, kerZod, “Tovs” da “wvims” da maTgan nawarmoebi ori uaryofiTi winadadeba da erTi pirobiTi winadadeba.

movniSnoT sabazo winadadebebi propoziciuri cvladebiT Sesabamisad p da

q. maSin Cveni argumenti Semdeg saxes miiRebs:

p⊃¬q q ¬p

mivceT am arguments CvenTvis ufro Cveuli saxe, Semdegi mosazrebebidan

gamomdinare: is rac weria, sinamdvileSi niSnavs imas, rom Tu WeSmaritia

p⊃¬q da WeSmaritia q maSin WeSmaritia ¬p, rac niSnavs ((p⊃¬q)∧q)⊃¬p) winadadebis WeSmaritebas. marTlac, is rom es tavtologiaa, advili

sanaxavia im meTodebiTac, romelic ukve viciT. Tumca, Cven zemoT

aRvniSneT, rom es xdeba zogjer da ara yovelTvis. Ees Tu marTlac asea,

maSin propoziciuri logika sakmarisi ar yofila imisaTvis, rom Cven mier

gamoTqmuli yvela argumenti aRvweroT. Ees ki seriozuli naklia, da unda

vcadoT, rom movxsnaT. marTlac, pirveli magaliTis analogiiT,

ganvixiloT Semdegi argumentacia:

am universitetis yvela studenti swavlobs logikas

giorgi am universitetis studentia

maSasadame, giorgi swavlobs logikas.

44 miuxedavad imisa, rom argumenti koreqtuli Cans, Cven gagviWirdeba misi

koreqtuloba propoziciuri logikis saSualebebiT avsaxoT. marTlac, am

argumentSi sami sabazo winadadebaa:

universitetis yvela studenti swavlobs logikas,

giorgi universitetis studentia,

giorgi swavlobs logikas.

avirCioT maTTvis propoziciuri cvladebi, vTqvaT, p, q da r. magram, am

SemTxvevaSi, Tu ki wina magaliTis analogias gavyvebiT, winadadeba p∧q⊃r, zogadad, ar aris WeSmariti, rac eWvis qveS ayenebs, Cvens, erTi SexedviT,

ueWvel argumentacias, rom rac samarTliania yvelasTvis, samarTliani

unda iyos erT-erTisTvisac. sxva sityvebiT, propoziciuri logika

sakmarisi ar yofila imisaTvis, rom Cven mier gamoTqmuli yvela argumenti

aRvweroT da, maSasadame, unda vnaxoT gza, misi saSualebebis

gaZlierebisaTvis. uxeSad rom vTqvaT, swored es aris predikatTa logikis

mizani da daniSnuleba. Tu rogor aris es SesaZlebeli, momdevno

paragrafebSi vnaxavT.

$2. sawinadadebo formebi, predikatebi, cvladebi da konstantebi.

$2.1 martivi winadadebis struqtura. winadadebaTa logikaSi, romelic

aqamde Cveni msjelobis ZiriTad Sinaars Seadgenda, martiv winadadebas,

erTian, daunawevrebel mTlianobad ganvixilavdiT. es garkveuli

mosazrebiT iyo nakarnaxevi. kerZod, Cven vcdilobdiT, erTis mxriv,

sawinadadebo makavSireblebis roli warmogveCina, xolo meores mxriv,

srulad gvenaxa rTuli winadadebebis struqturidan gamomdinare

logikisaTvis saintereso mimarTebebi da SeZlebisdagvarad aRgvewera isini.

axla, analogiuri amocana martiv winadadebebTan mimarTebaSi dgeba. ras

warmoadgens martivi winadadeba struqturulad? ra komponentebisagan

Sedgeba? ra logikuri mimarTebebi SeiZleba damyardes martivi winadadebis

komponentebs, anda Tavad martiv winadadebes Soris, Tu ki maT struqturas

daveyrdnobiT? da bolos, ramdenadaa SesaZlebeli am mimarTebaTa erTgvari

ganzogadeba im azriT, rom maTi formulebis saxiT warmodgena SevZloT da

ras mogvcems es?

cnobilia, rom sametyvelo enis nebismieri winadadeba, romelic raime

viTarebas aRwers (TxrobiTi winadadeba), sul cota, or komponents mainc

Seicavs: esenia is, rasac aRvwerT (rogorc momqmeds, anda moqmedebis

obieqts, da rasac gramatikaSi qvemdebares eZaxian), da is, risi

saSualebiTac mis, anda masze ganxorcielebul moqmedebas aRvwerT (da

rasac Semasmenels uwodeben). metic, rodesac gvaqvs qvemdebare da

Semasmeneli, amboben, rom viTareba praqtikulad aRwerilia da rom

45 sxvaMsityvis, anda sityvaTa jgufis damateba, mxolod ~aferadebs~ Sinaars.

amgvarad, SegviZlia vTqvaT, rom martivi TxrobiTi winadadebis struqturis

RerZi aris qvemdebare-Semasmeneli . sxva sityvebiT, aq gvaqvs is, razedac

raime gamoiTqmis da isic, rac gamoiTqmis. pirveli miuTiTebs sagans, xolo

meore, am sagnis daxasiaTebas iZleva. magliTad,

1. „bucefali“ cxenis saxelia

2. am saklaso oTaxis merxebi dazianebulia

3. dedamiwa planetaa.

zogadad, ra Sinaarsis matarebelia TxrobiTi winadadeba? yovel TxrobiT

winadadebaSi dadasturebulia an uaryofilia rom esa da es sagnebi aris

aseTi da aseTi. Cven SemTxvevaSi, magaliTad, vadasturebT, rom saxeli

„bucefali“ hqvia cxens, rom saklaso oTaxSi myofi merxebi aris

dazianebuli, xolo dedamiwa aris erT-erTi planeta. sxva sityvebiT,

winadadebaSi sagans (sagnebs) miewereba garkveuli niSan-Tvisebebi, anda

uariyofa, rom maT es niSan-Tvisebebi gaaCnia. magaliTad, Tu vityodiT, rom

4. dedamiwa ar aris planeta

marTalia, mcdar winadadebas gamovTqvamdiT, magram mainc winadadebas

gamovTqvamdiT, romelSic mcdarad aris uaryofili is garemoeba, rom

dedamiwa aris planeta. zogjer, TxrobiTi winadadebani ara saganTa

daxasiaTebas, aramed maT Soris damokidebulebas aRwers. magaliTad,

5. vano da kote mezoblebia

6. Jirafi spiloze maRalia

7. Zalosani asaniZe ufro Ronieria, vidre CogburTeli Sarapova, da a.S.

sagnebi, romlTa Sesaxebac winadadebas gamovTqvamT, sxvadsxva bunebisaa:

sulieri arsebani, usulo, warmosaxviTi, abstraqtuli. yvelaferi, rasac

SesaZloa Cveni azrovnebis Tvalsawieri gawvdes, rac SeiZleba Cveni

msjelobis obieqtad iqces da razedac, saerTod, SegviZlia raime

gamovTqvaT, warmoadgens sagans, romelic SeiZleba garkveuli sizustiT

aRiweros, da es aRweraA TxrobiTi winadadebiT gamoiTqvas. magram

aRniSnuli aRwera rom azriani gamogvivides, Cven, sul cota, ori tipis

obieqtebi gvWirdeba enaSi: saxelebi sagnebisaTvis da saxelebi im niSan-

TvisebebisaTvis, an damokidebulebisTvis, romlebsac am sagnebis mimarT

gamovTqvamT. Ppirvels, logikaSi termebs, xolo meores – predikatebs

veZaxiT. swored am ori tipis gamosaxulebebia, romlebic predikatTa

logikis asagebad ZiriTad „samSeneblo“ masalas warmoadgens.

46 §2.2. predikatebi, cvladebi da konstantebi. davubrundeT Cvens magaliTebs

da vcadoT gaverkveT Tu ras warmoadgens predikati? iqidan gamomdinare,

rac vTqviT, bunebrivia vifiqroT, rom mocemuli winadadebebidan, Tu

saxelebis aRmniSvnel sityvebs amovagdebT, rac dagvrCeba, swored

predikatic is unda iyos. am gzas Tu gavyvebiT, gveqneba Semdegi

――― cxenis saxelia

――― dazianebulia

――― planetaa.

―――― da ―――― mezoblebia

―――― maRalia vidre ―――――

―――― ufro Ronieria, vidre ――――

―――― (ar) aris planeta

Tu kargad davkvirdebiT, vnaxavT, rom aRniSnuli gamosaxulebebi

winadadebebad iqceva maSinaTve, Tu ki cariel adgilebs nebismieri

saxelebiT, da ara mxolod Cven magaliTebSi motanili saxelebiT,

SevavsebT. zogierTi gamosaxulebis winadadebad qcevas ki erTi saxeli ar

yofnis: oria saWiro. is, rom aseTi ram SesaZlebelia (anu am

gamosaxulebaTa winadadebebad gardaqmnaa SesaZlebeli), migviTiTebs imaze,

rom saqme, sinamdvileSi, sawinadadebo formebTan gvaqvs. Tumca es formebi

arsebiTad gansxvavdeba im formebisgan, romlebic propoziciur logikaSi

gvxdeboda. iq, sawinadadebo cvladebi winadadebebiT unda Segvecvala, rom

winadadeba migveRo. Aaq ki, Tu cariel adgilebs cvladebad aRviqvamT,

imisaTvis, rom azriani winadadeba miviRoT, konkretul obieqtTa

konkretuli saxelebi unda vixmaroT. maSasadame, cvladebi, Tu maT vixmarT,

Tavisi bunebiT, imave yaidisaa, rogorsac maTematikaSi vxvdebodiT,

rodesac, magaliTad vwerdiT x≥5, anda x≠y. Tumca es ar unda iwvevdes gakvirvebas, vinaidan „metia an toli“, anda „ar aris igive, rac“ iseTive

predikatebia, rogorc sxva danarCenebi. Uubralod, am bolo SemTxvevaSi, am

pedikatTa azrian winadadebebad gardaqmnisaTvis x da y ricxvTa RerZze unda iyos gansazRvruli, xolo rodesac, magaliTad, vambobT, rom viRac sxvaze

ufro Ronieria, es sxva da es viRac, cocxal arsebaTa samyaros unda

ekuTvnodnen.

imisaTvis, rom Cveni msjeloba ufro sagnobrivi gavxadoT, SevTanxmdeT

Semdegze:

SeTanxmeba 2.1. U iyos universumi, romelsac gansjis, an bWobis universums

vuwodebT, da yvela predikati, romelsac movixmarT, am universumze iyos

gansazRvruli im azriT, rom SegviZlia Cveni predikatebiT aRniSnuli

Tvisebebi da damokidebulebebi azrianad mivuyenoT universumis obieqtebs.

47 bWobis universumis obieqtTa saxeldebisaTvis vixmaroT laTinuri anbanis

dasawyisi mcire asoebi indeqsebiT, an maT gareSe, da maT erTeulovani

saxelebi, an konstantebi vuwodoT. magaliTad, konstantebia a,b, c,. . . a1,b1,c1. . . universumis obieqtTa nebismieri miTiTebisaTvis vixmaroT mcire asoebi

imave anbanis bolodan, indeqsebiT an maT gareSe, da maT cvladebi

vuwodoT. maSasadame cvladebi iqneba x,y,z,. . . x1, y1, z1,. . . predikatebisTvis vixmaroT amave anbanis Suidan mTavruli asoebi, indeqsebiT an maT gareSe,

romlis gverdiT frCxilebSi miTiTebuli iqneba cvladebis is raodenoba,

romelTa Senacvlebacaa aucilebeli konstantebiT, rom es predikatiani

gamosaxuleba azrian winadadebad gardaiqmnas. magaliTad, sapredikato

gamosaxulebebi iqneboda P(x), P1(x,y), Q(a,x) da a.S. sapredikato gamosaxulebaSi konstantebiT Sesanacvlebel cvladTa raodenoba am

gamosaxulebis e.w. adgilianobas (zogjer vityviT, predikatis

adgilianobas) gansazRvravs. magaliTad, Cven SemTxvevaSi, P erTadgiliani

predikatia, xolo P1 – ori. aucileblad mivaqcioT yuradReba imas, rom

gamosaxuleba Q(a,x) erTadgilania, radgan mis winadadebad gardaqmnas erTi

cvladis Senacvleba sWirdeba. Tumca, rogorc winadadeba, is raRac

mimarTebis gamomTqmelia P1 – is msgavsad. erTadgilian predikatebs xSirad

uwodeben Tvisebebs, xolo or da metadgilian predikatebs - mimarTebebs.

magaliTi 1. U iyos adamianTa simravle. Kkonsatanta a aRniSnavdes vanos,

xolo konstanta b – kotes. P(x,y) iyos adamianTa simravleze gansazRvruli

predikati „―――― mezobelia ――――“. maSin gamosaxuleba P(a,b) realurad

warmoadgens Cven mier zemoT motanili winadadebebidan erT-erTs, kerZod,

(5)-s. xolo gamosaxulebani P(a,x) da P(x,b) Sesabamisad ikTxeba rogorc „mezobelia vanosi“ da „mezobelia kotesi“. Bbunebrivia, isini gansxvavdebian

rogorc erTmaneTisgan, aseve sawyisi oradgiliani predikatisganac.

magaliTi 2. U iyos naturalur ricxvTa simravle. konstanta a aRniSnavdes

7-s, xolo konstanta b aRniSnavdes 3-s. P(x) iyos predikati x≥5, anu, Tviseba

„iyos xuTze meti“. maSin gamosaxuleba P(a) realurad aRniSnavs mcdar

winadadebas, xolo gamosaxulebani P(b) – WeSmarits.

$3. kavantorebi da kvantorebian winadadebaTa WeSmaritebiTi mniSvnelobebi.

cvladebis nacvlad sawinadadebo formaSi erTeulovani terminis Casma,

amgvarad, dasrulebul winadadebas gvaZlevs. Tumca arsebobs sxva gzac,

rasac kvantorebis Semotanas ukavSireben. Cveulebriv sityvaxmarebaSi

kvantorebi Seesatyviseba iseT frazebs, rogoricaa yvela da zogierTi.

48 marTlac, Tu gvaqvs sawinadadebo forma, magaliTad, ~x ≥ 5~, Semdegi ori gamosaxuleba ukve winadadebas warmoadgens:

a) yvela x-Tvis x ≥ 5

b) zogierTi x-Tvis x ≥5

amaTgan pirveli mcdaria, xolo meore WeSmariti. pirvels zogadobis

kvantori hqvia, xolo meores arsebobis. maTi aRniSvnisaTvis ixmareba

simboloebi ∀ da ∃ im cvladis miTiTebiT, romelsac es kvantorebi exeba.

Tu sawinadadebo formaSi erTze meti sxvadasxva cvladi gvxdeba, maSin mis

winadadebad gardasaqmnelad, Tu ki maT konstantiT ar vcvliT, imdeni

kvantori gvWirdeba, ramdeni sxvadasva cvladicaa.

cvladebi, romelTa mimarTac kvantori gamoiyeneba, bmuli cvladi ewodeba,

xolo ukvantoro cvlads - Tavisufali. amrigad, sawinadadebo forma erT

Tavisufal cvlads mainc Seicavs.

zogadobis kvantoris Semcvel winadadebas zogadi winadadeba hqvia. xolo

arsebobis kvantoris Semcvels - kerZoobiTi. ganvsazRvroT maTi

WeSmaritebis pirobebi:

zogadi winadadeba ∀x A(x) (ikiTxeba rogorc ~yoveli x iseTia, rom

WeSmaritia A(x)~ ) WeSmaritia maSin da mxolod maSin rodesac A(x)

sawinadadebo formas akmayofilebs x cvladis yoveli mniSvneloba gansjis

universumidan.

kerZoobiTi winadadeba ∃ x A(x) (ikiTxeba rogorc ~arsebobs x iseTi, rom

WeSmaritia A(x)~ ) WeSmaritia maSin da mxolod maSin rodesac A(x)

sawinadadebo formas akmayofilebs x cvladis erTi mainc (zogierTi)

mniSvneloba gansjis universumidan.

rodesac sawinadadebo forma erTze met cvlads Seicavs, maSin kvantorebi

SesaZloa TavsarTis saxiT dajgufdnen. Zalian mniSvnelovania Sereul

kvantorTa wyvilebi. aseTi wyvilebi orgvaria, da isini kargad unda

gavansxvavoT erTmaneTisgan;

• ∀x∃y niSnavs ~yoveli x-Tvis arsebobs y iseTi, rom~ da

• ∃x∀y niSnavs ~arsebobs iseTi x, rom yoveli y-Tvis~, rac

sazogadod erTi da igive ar aris.

kvantorebis Semotana mZlavr gamomsaxvelobiT saSualebebs iZleva,

imisaTvis rom sxvadsxva saxis msjelobaTa met-naklebad formalizebuli

analizi SevZloT.

49

magaliTi 3. martiv winadadebas, romelSic or predikats Soris kavSiri aris

gamoTqmuli, martivi kategoriuli winadadeba ewodeba. Teoria, romelic

aseT winadadebebs aRwers, logikisaTvis aristoteles droidanaa cnobili

da, realurad, logikis mecnierebisa da logikis swavlebis safuZvels

warmoadgenda

martivi kategoriuli winadadebebia:

a) vardi yvavilia; b) zogierTi skami gatexilia; c) adamiani mokvdavia da a.S.

im terminebidan, romlebic martivi kategoriuli winadadebis RerZs

warmoadgens, erTs, razedac azrs gamovTqvamT, vuwodebT subieqts, xolo

meores vuwodebT predikats. subieqti da predikati, erToblivad, martivi

kategoriuli winadadebis terminebs warmoadgens. mivaqcioT yuradReba imas,

rom sityva predikatis xmareba, rodesac martivi kategoriuli winadadebebze

vsaubrobT, gansxvavebulia, imisagan, rac Cven davadgineT. predikati aq

terminis ufro predikatis poziciaze yofnas niSnavs. amis gamo, erTi da

igive termini, SesaZloa, erT winadadebaSi predikatis adgilas vixiloT,

xolo meore SemTxvevaSi – subieqtisa. aseT viTarebaSi, pirvel winadadebaSi

is predikati iqneba, xolo meore winadadebaSi – subieqti. Tumca, orive

predikatia ($2.2)-Si dadgenili azriT. terminebis aRrevis aq ar unda

SegveSindes, vinaidan, sazogadod, moZRvrebas martivi kategoriuli

winadadebebis Sesaxeb predikatTa logikisgan damoukideblad ganixilaven.

Cveni magaliTis daniSnuleba ki isaa, rom vaCvenoT am ori Teoriis

urTierTkavSiris SesaZlebloba. kerZod is, rom predikatTa logikis

gamomsaxvelobiTi saSualebebi iZleva SesaZleblobas martivi kategoriuli

winadadebebis gamoTqmisa. Tu rogor xdeba es, amas qvemoT vnaxavT. zemoT

motanil winadadebebSi terminebia ~vardi~, ~yvavili~, ~skami~, ~gatexili~,

~adamiani~ da ~mokvdavi~. aqedan, ~vardi~, ~skami~ da ~adamiani~, mocemul

winadadebebSi, subieqtebia, xolo ~yvavili~, ~gatexili~ da ~mokvdavi~ ki

predikatebi. martiv kategoriul winadadebaSi terminebs Soris kavSirebi

SesaZloa gansxvavdebodes erTmaneTisagan Tvisebrivad da raodenobrivad:

raodenobriobis mixedviT martivi kategoriuli winadadeba iyofa orad

(zogadi da kerZo), romelTagan TiToeuli or-orad iyofa ukve

Tvisobriobis mixedviT, dadebiTad, an uaryofiTad. ase rom gvaqvs

i. zogad-dadebiTi, ii. zogad-uaryofiTi,

iii. kerZoobiT-dadebiTi, da iv. kerZoobiT-uaryofiTi martivi kategoriuli winadadebebi yuradReba mivaqcioT im garemoebas, rom struqturulad martivi

kategoriuli winadadebebi orgvari elementebisagan Sedgeba: terminebi, anu

subieqti da predikati, da is, rac am terminebis azrianad dakavSirebis

SesaZleblobas iZleva, is rac gamoiTqmis rogorc aris, ar aris an yvela

50

aris, arc erTi ar aris, zogierTi aris, zogierTi ar aris. es niSnavs, rom

erTi da igive terminebiT SegviZlia avagoT sxvadasxva formis martivi

kategoriuli winadadeba.

magaliTad terminebidan adamiani da mokvdavi SegviZlia avagoT

• zogad-dadebiTi : yvela adamiani aris mokvdavi

• zogad-uaryofiTi: arc erTi adamiani ar aris mokvdavi • kerZoobiT-dadebiTi: zogierTi adamiani aris mokvdavi • kerZoobiT-uaryofiTi: zogierTi adamiani ar aris mokvdavi zogadad, Tu S da P Seabamisad aRniSnavs subieqtsa da predikats, xolo a,

e, i da o aRniSnavs zogad-dadebiT, zogad-uaryofiT, kerZoobiT-dadebiT da

kerZoobiT-uaryofiT kavSirs, gveqneba Semdegi sqema:

martivi kategoriuli winadadebis saxeoba

simboluri Canaweri simboluri Canaweris wakiTva

zogad-dadebiTi SaP yvela S aris P zogad-uaryofiTi SeP arc erTi S ar aris P kerZoobiT-dadebiTi SiP zogierTi S aris P kerZoobiT-uaryofiTi SoP zogierTi S ar aris P

sinamdvileSi, es aris martivi kategoriuli winadadebebis tradiciuli Caweris forma, romelic saukuneebis manZilze moixmareba logikaSi, da bevri saxelmZRvanelo dResac am Canawerebs iyenebs. Tumca, logikaSi kvantorebis SemoRebis Semdeg, es Canawerebi nel-nela xmarebidan gamodis, vinaidan maTi Canacvleba advilad SesaZlebelia kvantorebiani formulebiT. martivi kategoriuli winadadebebi SaP. SeP, SiP da SoP kvantorebis gamoyenebiT Caiwereba Semdegnairad:

SaP ∀x(S(x)⊃P(x)) SeP ∀x(S(x) ⊃¬P(x)) SiP ∃x(S(x) ∧ P(x)) SoP ∃x(S(x)∧¬ P(x))

§3.1. kvantorebiani formulebis Tvisebebi. qvemoT mocemulia im formulaTa

CamonaTvali, romelTa codna amartivebs kvantorebian formulebTan

muSaobas im TvalsazrisiT, rom gvaCvenebs kvantorebis Semcveli

formulebis urTierTdamokidebulebas, maT mimarTebas propoziciur

makavSireblebTan da gviCvenebs im gzas, romelsac erTi formulidan

SeuZlia meoremde migviyvanos. maT zogjer predikatTa logikis kanonebs

uwodeben

51

∀xA(x) ⊃ A(a)

zogadobis kvantoris mocilebis wesi

A(a) ⊃ ∃xA(x)

arsemobis kvantoris SemoRebis wesi

∀xA(x) ⊃ ∃xA(x)

kvantorTa daqvemdebarebis wesi

¬∀xA(x) ≡ ∃x¬A(x)

¬∃xA(x) ≡ ∀x¬A(x)

de morganis kanonebi kvantorebisaTvis

∀x(A(x) ∧ B(x)) ≡ ∀xA(x) ∧ ∀xB(x) ∃x(A(x) ∨ B(x)) ≡ ∃xA(x) ∨ ∃xB(x) ∃x(A(x) ∧ B(x)) ⊃ ∃xA(x) ∧ ∃xB(x) ∀xA(x) ∨ ∀x B(x) ⊃ ∀x(A(x) ∨ B(x))

distribuciis wesebi kvantorebisaTvis

∀x∀yR(xy) ≡ ∀y∀xR(xy) ∃x∃yR(xy) ≡ ∃y∃xR(xy) ∀x(A(x) ⊃B(x)) ≡ ¬∃x (A(x) ∧¬ B(x))

eqvivalentobis wesebi

∃x∀yR(xy) ⊃ ∀y∃xR(xy) ∀x(A(x) ⊃B(x)) ⊃ (∀xA(x) ⊃ ∀x B(x) )

∀xyA(x,y) ⊃ ∀zA(z,z)

implikaciis wesebi

aqac, aseve kargi iqneba, Tu am formulebis SinaarsSi gaverkveviT, da im

principebs davimaxsovrebT, rac maTi saSualebiTaa gamoTqmuli.

52 $3.2. predikatebi da simravleebi. vTqvaT, U aris naturalur ricxvTa

simravle da is Cveni gansjis universums warmoadgens. xolo P(x), Q(x) da R(x) aris Semdegi predikatebi

P(x) iyos (x≥25)

Q(x) iyos (x+7=25)

R(x) iyos (x<25)

advili sanaxavia, rom TiToeuli maTgani garkveuli ricxvebisaTvis

WeSmaritia, ხოლო სხვა რიცხვებისათვის - არა. მაგალითად, P(x) ჭეშმარიტია 25-ს ზემოთ ყველა რიცხვისთვიs, თავად 25-ის ჩათვლით, და ასეთ რიცხვთა სიმრავლე უსასრულოა. Q(x) ჭეშმარიტია ერთადერთი რიცხვისათვის და ეს რიცხვი არის 18. რაც შეეხება R(x)-ს, ის ჭეშმარიტია 1-დან 24-ის ჩათვლით რიცხვებისათვის.

შევთანხმდეთ შემდეგზე: D-ze gansazRvruli, რაიმე S(x) predikatis WeSmaritebis are vuwodoT D-s elementTa im qvesimravles, romlebiTvisac

S(x) predikati WeSmaritia. simartivisaTvis, da imis mimaniSneblad, rom

mocemuli simravle S(x) predikatis WeSmaritebiTi area, aRvniSnoT is S-iT. magaliTad, Cven SemTxvevaSi, gveqneba

P(x) predikatis WeSmaritebiTi area U simravlis usasrulo qvesimravle P = {x:

x≥25},

Q(x) predikatis WeSmaritebiTi area U simravlis erTeulovani qvesimravle

Q =Q{18}, xolo

R(x) predikatis WeSmaritebiTi area U simravlis sasruli qvesimravle R = { x:

1≤x<25}.

vinaidan simravleebi raRac korelaciaSi gamodian predikatebTan,

saintereso iqneboda gvenaxa, xom ar arsebobs raime kavSiri simravlis

Teoriis kanonebsa da predikatTa logikis debulebebs Soris? gavsinjoT es

SesaZlebloba magaliTze:

aviRoT simravleTa Teoriis, raime WeSmariti debuleba, vTqvaT,

(1). (A⋃B)′ = (A′⋂B′)

da vnaxoT, ra iqneboda misi Sesatyvisi formula predikatTa logikaSi.

amisaTvis, simravlis damatebas SevxedoT rogorc uaryofas, gaerTianebas,

rogorc disiunqcias, kveTas, rogorc koniunqcias, xolo is garemoeba, rom

zemoT motanili debuleba ZalaSia nebismieri simravlisaTvis mocemuli

53 universumidan (Tu ki am universumis elementebi ekuTvnis an erT, am meore

simravles), gamovxatoT zogadobis kvantoris meSveobiT. maSin gveqneba

(2). ∀x(¬(A(x)∧B(x)) ≡ (¬ A(x) ∨¬ B(x))).

Zalian hgavs imas, rom (1) WeSmaritia maSin, rodesac WeSmaritia (2), da

piriqiT. vnaxoT, marTlac ase aris Tu ara? ganvixiloT pirveli varianti:

viciT, rom (1) aris WeSmariti. sakiTxi davsvaT ase: SesaZlebelia Tu ara

am pirobebSi (2) iyos mcdari? vTqvaT, rom SesaZlebelia. magram ras niSnavs

(2)-s mcdaroba? Ees niSnavs, rom WeSmaritia misi uaryofa. Ees ki, Tavis

mxriv, niSnavs imas, rom Cvens universumSi arsebobs erTi mainc iseTi

obieqti, vTqvaT, a, romlisTvisac samarTliania

¬(A(a)∧B(a)) ≢ (¬ A(a) ∨¬ B(a)),

sxva sityvebiT, ori ¬(A(a)∧B(a)) da (¬ A(a) ∨¬ B(a)) debulebidan arc orivea

erTdroulad WeSmariti, da arc orivea erTdroulad mcdari. maSasadame,

Tu, am SemTxvevaSi, WeSmaritia ¬(A(a)∧B(a)), maSin mcdari unda iyos (¬ A(a) ∨¬ B(a)), da piriqiT. vTqvaT, marTlac asea. magram pirvelis WeSmariteba

niSnavs, rom a∉A, an a∉B. axla SevxedoT ras niSnavs, rom mcdaria ¬ A(a) ∨¬ B(a)? didi gamWriaxoba ar Wirdeba imis danaxvas, rom es unda niSnavdes Semdegs: a∈A da a∈B. magram erTsa da imave elementisTvis SeuZlebelia

erTdrulad is kidec ekuTvnodes raimes da kidec ar ekuTvnodes. miviReT

winaaRmdegoba. maSasadame es gza ar varga. maSin, iqneb meore debulebaa

WeSmariti da pirveli mcdari? zusted iseve SegviZlia vaCveniT, rom es

SemTxvevac SeuZlebel Sedegs gvaZlevs. Tumca am winaaRmdegobamde

(SeuZlebel Sedegebamde) Cven imis daSveebam migviyvana, rom (1) WeSmaritia,

xolo (2) mcdari. amitom es daSvebaa Tavad mcdari da uari unda vTqvaT

masze. Sedgad, miviRebT, rom Tu (1) WeSmaritia, (2) aucileblad WeSmaritia.

anlogiurad SegviZlia davrwmundeT imaSic, rom Tu (2) WeSmaritia, (1)

aucileblad WeSmaritia.

sxvaTaSoris, is, rom simravleTa Teoriis WeSmarit debulebebsa da

predikatTa logikis debulebeb Soris aseTi damikidebuleba arsebobs,

Zalian amartivebs zogierT Temaze msjelobas. garda amisa, is

SesaZleblobas iZleva predikatebis ganxilvisas venis diagramebi

gamoviyenoT.

savarjiSos saxiT, ganixileT SemTxveva, rodesac saqme oradgilian

predikatebs exeba. ZalaSi iqneba Tu ara maTTvisac igive?

54 damateba 1. cota ram dasabuTebis Sesaxeb

adamianebi, cxovrebis manZilze, xSirad cdiloben erTmaneTi daarwmunon da

amis uamravi mizezi gaaCniaT: magaliTad, politikuri moRvaweebi cdiloben

daarwmunon sxvebi maT mier arCeuli politikuri kursis momgebianobaSi,

rom arCevnebisTvis mosaxleobis mxardaWera moipovon. sasamarTlo

procesze, prokurori cdilobs mosamarTlis darwmunebas, rom

sabraldeblo skamze myofi marTlac damnaSavea, xolo advokati, iqve, imave

mosamarTles arwmunebs imaSi, rom saqme sapirispirodaa warmodgenili.

mecnierebi davoben xSirad erTsa da imave monacemTa interpretirebisas da

es dava sxva araferia, Tu ara erTmaneTis darwmunebis mcdeloba TavianTi

midgomis sisworeSi. uamravi msgavsi magaliTis motana SeiZleba. nebismier

situaciaSi, rodesac adamianebi azrs cvlian, msjeloben, davoben, siaxles

nergaven da sxv. urTierTdarwmunebis mcdelobasTan gvaqvs saqme.

sazogadod, yvela am SemTxvevaSi, erTsa da imave sqemas mimarTaven xolme:

jer TavianT mosazrebas gamoTqvamen, xolo Semdeg cdiloben garkveuli,

TavianTi simarTlis damadasturebeli argumentebis moSveliebas, rom es

mosazreba damajerebeli, anu sarwmuno gaxadon. argumentebi, Cveulebriv,

mravalgvaria da damokidebulia rogorc msmenelisa da mosaubris

inteleqtze, aseve im sagnis raobaze, romelic gansjis sababi gamxdara.

xSirad, da praqtikulad umetes SemTxvevaSi, argumenti demonstrirebiT

Semoifargleba: me gaCvenebT rames da es sakmarisi xdeba, rom Tqven

darwmundeT. Qam dros, qvecnoberad vgulisxmobT, rom adamiani sakuTar

mxedvelobas endoba da es asecaa. sxvaTa Soris,Aamitomacaa, rom zogjer

vxmarobT gamoTqmas “Tu Tvali ar matyuebs”. vgulisxmobT, rom, sazogadod,

Tvali ar gvatyuebs. marTalia, dasarwmuneblad adamianebi zogjer fizikur

zemoqmedebasac mimarTaven. magaliTad, rodesac bavSvi cudi saqcielis gamo

isjeba. magram, vinaidan fizikuri zemoqmedeba, zogadad, argumentad ar

aRiqmeba, amitom dasja metwilad aRzrdis araefeqtur meTodadaa miCneuli:

me gijereb ara imitom, rom Sen argumentirebulad damarwune imaSi, rom

rac gavakeTe cudia, aramed imitom, rom dRes Seni meSinia. bunebrivia,

rogorc ki es SiSi mTavrdeba, imave viTarebas vubrundebiT: me isev cudad

viqcevi. sxva sityvebiT, Sen ki ar damarwmune, Sen maiZule da zogadad, me

Seni ar mjera. ai es dgas fizikuri zemoqmedebis ukan da sinamdvileSi,

araviTar darwmunebasTan aq saqme ar gvaqvs. darwmuneba argumentis

moSveliebasTan aris asocirebuli da mTel am process Cven dasabuTebas

veZaxiT. davubrundeT argumentebis mravalgvarobas. demonstrireba,

sxvaTaSoris, ar gulisxmobs mxolod raimeze miTiTebas. xSirad, es sakmaod

rTuli proceduraa: magaliTad, rodesac fizikosi, an nebismieri

bunebaTmcodne, eqsperiments ayenebs, esec demonstrirebaa. aseve xSirad,

argumentaciisas, adamianebi msgavsi magaliTebis moSveliebas mimarTaven da

55 amas analogiiT dasabuTebas uwodeben. analogiebs, magaliTad sasamarTlo

praqtikaSi mimarTaven xolme, rodesac ganaCenis gamarTleba adre momxdari

msgavsi SemTxvevisTvis miRebuli gadawyvetilebiT surT. amas, zogadad,

precendentul samarTals eZaxian. sxva uamravi SemTxvevis aRwera SeiZleba,

rodesac yoveldRiurobaSi Cven sxvadasva arguments moviSveliebT xolme

da vfiqrobT, rom SevZeliT rogorc Cveni simarTlis damtkiceba, aseve

msmenelis darwmunebac. Tumca, xSirad, rac Cven argumenti gvgonia,

arguments saerTod ar warmoadgens, anda warmoadgens, magram is sakmarisi

ar aris, rom gamoTqmuli mosazreba sabolood davadasturoT. am mxriv,

sxvaTaSoris, yvelaze momgebianad, maTematika gamoiyureba. SemTxveviTi ar

aris iseTi gavrcelebuli gamonaTqvamebi, rogoricaa maTematikuri sizustiT, maTematikurad gaTvlili, maTematikurad dasabuTebuli, da sxv.

es gamonaTqvamebi, zogadad, matematikaSi gamoyenebuli dasabuTebis xerxebis

mimarT Cvens saerTo ndobaze miuTiTebs. ra xerxebia es? da SegviZlia Tu

ara maTi imdagvari aRwera, rom Cvens sakuTar argumentaciaSi aq

gamovlenili strategiebis gamoyeneba SevZloT?

pirvel rigSi SevTanxmdeT imaze, Tu ras davarqvaT dasabuTeba. CavTvaloT,

rom ena, romelsac movixmarT, aris propoziciuri logikis ena, romelic,

SesaZloa, gamdidrebuli iyos zogirTi simboloTi, rom winadadebaTa

erTobliobebze miTiTeba SevZloT. im warmodgenebidan gamomdinare, rac

Cven intuiciurad gagvaCnia, dasabuTeba warmoadgens garkveul jaWvs

winadadebebisa, romlebic TanamimdevrobiT, ase vTqvaT, gvaaxloeben Cvens

ZiriTad, dasasabuTebel winadadebasTan. bunebrivia, CavTvaloT, rom am

jaWvis bolo winadadeba unda iyos swored is, ris dasabuTebasac vlamobT.

maSasadame, raime tn formulis dasabuTeba t1, t2, t3, . . tn-1-dan aris t1, t2, t3, . . .tn-1,

tn winadadebaTa Tanamimdevroba, sadac TiToeuli ti<n, warmoadgens imas, rasac, Cveulebriv, wanamZRvrebs vuwodebT, da romlebic, Tavis mxriv

warmoadgenen an daSvebebs, ase vTqvaT, varaudebs, romlebsac dasabuTeba

eyrdnoba da romelTa WeSmariteba CvenTvis sarwmunoa, an CvenTvis ukve

cnobil (im azriT, rom maTi WeSmariteba dasabuTebuli gvaqvs)

WeSmaritebebs, an am ori tipis debulebebidan garkveuli wesebis mixedviT

miRebul axal debulebebs, vinaidan wesebi, romlebsac am dros veyrdnobiT,

iseTnairad gvaqvs SerCeuli, rom isini Tavdapirveli winadadebebis

WeSmaritebas gvinarCuneben. rodesac gvinda davweroT, rom t1, t2, t3, . . tn-1-dan

gamomdinareobs tn , vixmarT Semdeg Canawers: t1, t2, t3, . . .tn-1 ├ tn . davukvirdeT

Cveni Canaweris Sinaarss. Ees xom niSnavs imas, rom Tu WeSmaritebia t1, t2, t3, .

. .tn-1 formulebi, aseve WeSmariti unda iyos tn. anu, WeSmariti unda iyos

implikacia (t1∧ t2∧ t3∧ . . .∧tn-1⊃ tn). magram amis Semowmeba, sazogadod, xom WeSmaritebiTi cxrilebiTac SeiZleba? ra Tqma unda, SeiZleba, magram es

yovelTvis gamosavali ar aris da seriozul debulebaTa dasasabuTeblad

56 Cven mas TiTqmis ar mivmarTavT. imas, rac axla aRvwereT pirdapir

dasabuTebas uwodeben da is dasabuTebis erT-erTi yvelaze gavrcelebuli,

Tumca, metwilad, rTuli strategiaa.

pirdapiri dasabuTeba, ZiriTadad, eyrdnoba implikaciis tranzitulobis

Tvisebas, kerZod, Tu viciT, rom p⊃r da, amave dros, r⊃q, maSin SegviZlia

davaskvnaT, rom p⊃q. wesebi, romliTac, magaliTad, propoziciur logikaSi,

axal formulaze gadasvla xdeba, da rasac daskvnis wesebi hqvia,

metwilad,, oria. es aris e.w. modus ponens (sruli saxeli modus ponendo ponens-

konseqventis dadastureba (ponendo) antecedentis dadasturebis (ponens)

meTodiT (modus)) da eqvivalentur debulebaTa urTierT Canacvleba ukve

arsebul da WeSmaritad miCneul winadadebaSi. erTaderTi, rac daskvnis

wesebs moeTxoveba aris is, rom maTi gamoyeneba ar unda cvlides WeSmariti

debulebis WeSmaritebas (eqvivalentur debulebaTa urTierT Canacvlebis

SemTxvevaSi), da ori WeSmariti debulebidan mesameze gadasvla, aseve unda

inarCunebdes WeSmaritebas (modus ponens –is SemTxvevaSi). is, rom orive zemoxsenebuli wesi am pirobebs akmayofilebs, advili dasanaxia. marTlac,

Tu, vTqvaT, P(s) aris propoziciuri logikis raime rTuli formula, sadac

misi erT-erTi sawinadadebo komponenti, sulerTia rogori, aris s, maSin Tu

viciT, rom s≡q (viciT, niSnavs, rom am eqvivalentobis dasabuTeba

SegviZlia), da viciT, rom P(s) WeSmaritia, maSin asev WeSmariti unda iyos P

P(q), romelic pirvelisagan mxolod imiTaa gansxvavebuli, rom masSi s-is

nacvlad (yvelgan, an zogan) weria q. marTlac, s SeiZleba iyos 0, an SiZleba

iyos 1. rodesac is 0-ia, asTivea q-c, da am SemTxvevaSi, Tu P(s) WeSmaritia,

maSin aseve WeSmariti unda iyos P(q). analogiuri msjeloba ZalaSia

rodesac s aris 1. modus ponens –is SemTxvevaSi saqme ufro martivadaa da is ((p⊃q)∧p)⊃q -is tavtologiurobis SemowmebiT Semoifargleba.

arapirdapiri dasabuTeba. zemoT vaxseneT, rom pirdapiri dasabuTeba,

erTgvarad rTuli strategiaa. xSirad, pirdapiri dasabuTebis povna imdenad

rTuli amocanaa, rom metwilad, cdiloben, Tu SesaZlebelia, is Tavidan

aicilon da sxva, ufro ioli da intuiciisTvis ufro naTeli meTodebi

ipovon. aman gansazRvra dasabuTebis, ase vTqvaT, arapirdapiri meTodebis

Camoyalibeba rogorc logikaSi, aseve maTematikaSi. qvemoT, yuradRebas

SevaCerebT orze, romelic naTel warmodgenas gviqvmnis Tavad arapirdapiri

midgomis arsze.

kontrapoziciiT dasabuTeba eyrdnoba eqvivalentobas (p⊃q) ≡ (¬q ⊃ ¬p). Mmisi

arsi mmdgomareobs SemdegSi: imisaTvis rom davasabuToT, rom (p⊃q)

davuSvaT, rom mcdaria q da Tu miviReT am daSvebis safuZvelze, rom

mcdaria p, maSin, zemoTmotanili eqvivalenobis safuZvelze, SegviZlia

57 dasabuTebulad miviCnoT (p⊃q). erTaderTi problema, romelsac aq SeiZleba

gadavawydeT, aris sawyisi, dasasabuTebeli implikaciis swori

kontrapozicuri formis povna, rac gamoucdeli TvalisaTvis, zogjer sul

ar aris advili.

dasabuTeba winaaRmdegobamde miyvaniT. Aam meTodis laTinuri saxelia

reductio ad absurdum da dasabuTebis erT-erT uZveles gzas warmoadgens. misi

arsi SemdegSi mdgomareobs: (p⊃q)- s dasabuTebisTvis davuSvaT, rom

WeSmaritia p da WeSmaritia ¬q. axla, Tu Cven logikurad swori msjelobiT

mivediT iqamde, rom davasabuTeT raime s debulebis uaryofa, anu miviReT,

rom WeSmariti unda iyos ¬s, xolo s-is Sesaxeb ki manamde cnobili iyo, rom

is WeSmaritia, gamova rom s erTdroulad WeSmariticaa da mcdaric (mcdari

unda iyos, vinaidan SeSmaritia misi uaryofa). Ees ki absurdia, romlamdec

migviyvana Tavdapirvelma daSvebam, rom WeSmaritia p da WeSmaritia ¬q. maSasadame, Tavad daSvebaa mcdari da adgili unda hqondes mis uaryofas,

rac niSnavs, rom WeSmaritia (p⊃q). sxvaTa Soris, naTqvamSi advilad

davrwmundebiT, Tu propoziciuri logikidan gavixsenebT, ras niSnavs

koniunqciis uaryofa, ras niSnavs ormagi uaryofa da rogoraa SesaZlebeli

implikaciis eliminacia.

magaliTi 1.

imisaTvis, rom ukeT gavacnobieroT is, rac dasabuTebis Sesaxeb iTqva, vcadoT

propoziciuri logikis raime debulebis dasabuTeba orive meTodis gamoyenebiT.

vTqvaT, es iyos (p∧¬p)⊃q

1. pirdapiri damtkiceba: davazustoT, unda davuSvaT antecedenti da vcadoT

movideT konseqventamde(!)

1.1. (p∧ ¬p) daSveba

1.2. (p∧q⊃p) propoziciuli logikis tavtologia

1.3. (p∧¬p⊃p) (1.2)-Si q-s nacvlad ¬p

1.4. p modus ponens (1.1), (1.3) 1.5. (p∧q⊃q) propoziciuri logokis tavtologia

1.6. (p∧¬p⊃¬p) (1.5)-Si q-s nacvlad ¬p

1.7. ¬p modus ponens (1.1), (1.6)

1.8. q⊃(q∨p) propoziciuri logokis tavtologia

1.9. ¬p⊃(¬p∨q) (1.8)-Si q-s nacvlad ¬p p-s nacvlad q

1.10. ¬p∨q modus ponens (1.7), (1.9)

1.11. p⊃q (1.10)-dan implikaciis eliminacia da modus ponens

1.12. q (1.4), (1.11), modus ponens

58 2. iribi dasabuTeba. davazustoT: aq jer vpoulobT dasamtkicebeli debulebis

swor kontrapoziciur variants da Semdeg mas vasabuTebT. (p∧¬p)⊃q formulis

kontrapozicia iqneba ¬q⊃(¬(p∧¬p)). Mmagram de morganisa da ormagi uaryofis

kanonebis ZaliT am implikaciis konseqventi aris (¬p∨p). anu, Cveni

dasamtkicebeli debuleba Semdeg saxes miiRebs: ¬q⊃(¬p∨p). Aam ukanasknelis

dsabuTeba elementarulia (¬p∨p) formulis tavtologuirobis gamo.

damateba 2. savarjiSoebi

simravleebi

1. davamtkicoT, rom nebismieri A da B simravlisaTvis samarTliania

Ø ⊂ (A⋂B)⊆ (AUB)

SeniSvna: damtkicebisa gamoiyeneT damateba 1 –Si ganxiluli meTodebi.

2. vaCvenoT, rom nebismieri A, B da C simralisaTvis U U universaluri simravlidan,

adgili aqvs Semdegs:

2.1. AU (BUC) = (AUB)U C 2.1.´ A⋂(B⋂C) = (A⋂B)⋂C

2.2. AUB = BUA 2.2.´ A⋂B = B⋂A

2.3. AU(B⋂C) = (AUB)⋂∩(AUC) 2.3.´ A⋂(BUC) =(A⋂B)∪(A⋂C)

2.4. AUØ = A 2.4.´ A⋂U = A

2.5. AUA´ = U 2.5.´ A⋂A´ = Ø

SeniSvna: miaqcieT yuradReba, rom Cveneba, igive demonstrirebaa da

maSasadame, garkveuli azriT, dasabuTebac.

3. aageT venis diagramebi TiToeuli tolobisaTvis me-2 savarjiSodan.

4. CawereT is simravlisTeoriuli tolobebi, romlebic propoziciuri

logikis Semdeg kanonebs Seesabameba da daamtkiceT maTi marTebuloba venis

diagramebis saSualebiT.

(p∧q) ≡ (q∧p) (pvq) ≡ (qvp) ((p∧q)∧r)≡(p∧(q∧r)) ((pvq)vr)≡ (pv(qvr))

59

(p∧(qvr))≡ ((p∧q)v(p∧r)) (pv(q∧r)) ≡((pvq)∧(pvr)) ┐(p∧q)≡(┐p v ┐q)

propoziciuli logika

1. avagoT WeSmaritebiTi cxrilebi Semdegi formulebisaTvis

((p∨q)∨r) ≡(p∨(q∨r))

┐(p∧q)≡(┐p ∨ ┐q)

¬(p⊃q) ≡ (p ∧ ¬q)

(p⊃q) ≡ ((p ⊃q) ∧ ( q ⊃ p))

(p∧(q∨r)) ≡ ((p∧q)∨(p∧r))

2. WeSmaritebiTi cxrilebis saSualebiT davrwmundeT, rom (A∧P) ⊃Q maSin

da mxolod maSin, rodesac A⊃(P⊃Q), sadac A , P da Q nebismieri

formulebia propoziciuri logikidan

3. aqsiomaturi Teoria vuwodoT debulebaTa iseT erToblibas romelic

Semdeg pirobebs akmayofilebs:

vadgenT sawyisi teminebis srul sias (teminebi, romelTa gansazRvreba

saWiro ar aris, vinaidan maTi moxmareba konteqstebSi cxadad aris

miCneuli)

vadgenT sawyisi winadadebebis sias, anu winadadebebisas, romlebic

miCneulia WeSmaritebebad maTi gamamarTlebebli bWobis gareSe. maT

sxvanairad aqsiomebi hqvia.

yvela sxva termini, romelic SesaZloa gamoyenebuli iqnas Teoriis

debulebaTa formulirebisas sawyisi terminebiT unda iyos gansazRvruli.

yvela debuleba, romelic aqsiomaTa siaSi ar gvxvdeba, WeSmaritad aris

miCneuli mxolod mas Semdeg, Tu ki igi gamoyvanilia aqsiomebidan da ukve

dasabuTebuli(damtkicebuli) debulebebidan dsakvnis SeTanxmebuli

wesebis meSveobiT. aseT debulebebs mocemuli Teoriis Teoremebi hqvia.

vTqvaT gvaqvs aqsiomaturi Teoria, romlis ena Sedgeba propoziciuri

logikis sawyisi simboloebisa da maTgan agebuli gamosaxulebebigan,

xolo aqsiomebia (mendelsoni, 1976)

• p⊃(q⊃p)

• (p⊃(q⊃s))⊃(( p⊃q)⊃( p⊃s))

• ((¬q⊃¬p)⊃((¬ q⊃p)⊃q)

Ddaskvnis wesebad SevinarCunoT modus ponens, da Canacvleba. davamtkicoT,

rom Semdegi formulebi Teoremebia

((p∨q)∨r) ≡(p∨(q∨r))

┐(p∧q)≡(┐p ∨ ┐q)

60

¬(p⊃q) ≡ (p ∧ ¬q)

(p⊃q) ≡ ((p ⊃q) ∧ ( q ⊃ p))

(p∧(q∨r)) ≡ ((p∧q)∨(p∧r))

4. A debulebaze vityviT, rom is logikurad gamomdinareobs B –dan,

rodesac implikacia B⊃A aris tavtologia. vaCvenoT, rom Semdegi debulebebi ZalaSia: 4.1. igiveobis wesi: yoveli winadadebidan logikurad Tavad es windadeba

gamomdinareobs.

4.2. wanamZRvarTa Camatebis wesi. Tu wanamZRvarTa garkveuli erTobliobidan

gamomdinareobs raime danaskvi, maSin rogorc ar unda gavafarTovoT

wanamZRvarTa wre, es danaskvi ZalaSi rCeba.

4.3. wanamZRvarTa moklebis wesi. Warbi gameorebebis amoReba wanamZRvaTa

simravlidan gamomdinareobas ar cvlis.

4.4. gadanacvlebis wesi. wanamZRvarTa Tanamimdevrobis Secvla

gamomdinareobas ar cvlis

4.5. logikuri gamomdinareobis tranzitulobis wesi. wanamZRvarTa Sedegebis

Sedegebi TviTon wanamZRvarTa Sedegebia

miTiTeba: TiToeuli winadadeba gamoxateT propoziciuri logikis enaze da

aCveneT, rom miRebuli formulebi tavtologiebia.

predikatTa logika

Tu ki propoziuri logikis daskvnis wesebs davumatebT wesebs

kvantorebisTvisac, SesaZlebeli gaxdeba Teoremebis damticeba predikatTa

logikisTvisac. magaliTad, Tu daskvnis wesad miviRebT

B ⊃ A(x)

(K) -----------------------

B ⊃ ∀xA(x)

sadac B ar Seicavs x-s Tavisuflad, advilad SegviZlia vaCvenoT e.w. ganzogadebis

wesis marTebuloba, romelic gulisxmobs, rom Tu WeSmaritia A(x) , maSin is

WeSmaritia yoveli x-Tvis, sxva sityvebiT, WeSmaritia ∀xA(x). vnaxoT, rogor aris es

SesaZlebeli:

1. A(x) daSveba 2. (p∨ ¬p) ⊃ A(x) daSvebdan winadadebaTa logikis kanonis gamoyenebiT da

modus ponens-iT. konkretulad viyenebT p⊃(q⊃p), sadac p-Si vanacvlebT A(x), xolo

q-Si (p∨¬p)-s da Semdeg viyenebT modus ponens-s

3. (p∨ ¬p) ⊃∀xA(x) (2)-dan (K)-s ZaliT

4. p∨ ¬p winadadebaTa logikis kanoni

5. ∀xA(x) (3)-dan da (4)-dan modus ponens-iT

61

Tu gavixsenebT, Tu rogor ganvsazRvreT dasabuTeba, davrwmundebiT, rom Cven

mier warmodgenili (1)-(5) winadadebebis Tanmimdevroba aris (5) winadadebis

damtkiceba A(x) wanamZRvridan.

5.1.1.1. ganzogadebis wesis gamoyenebiT davamtkicoT Teorema

∀x(A(x)∨¬ A(x)) 5.1.1.2. zogadobis kvantoris mocilebis wesis gamoyenebiT davamtkicoT

Teorema ∀x(A∨B(x)) ⊃(A∨∀xB(x)) 5.1.1.3. kvantorTa ra wesebis gamoyeneba dagvWirdeba Semdegi

formulebis dasamtkiceblad?

• ∀xA(x) ≡ ¬∃x¬ A(x)

• ∀x(A(x) ⊃B(x)) ⊃ (∀xA(x) ⊃ ∀x B(x) ) • ∀x(A(x) ⊃B(x)) ≡ ¬∃x (A(x) ∨ ¬ B(x))

5.1.1.4. logikuri kvadrati:

i. or martiv kategoriul winadadebaze vityviT, rom isini erTmaneTis

kontradiqtorulia, Tu ki SeuZlebelia, rom orive erTdrouliad iyos WeSmariti,

da orive erTdroulad iyos mcdari.

ii. or martiv kategoriul winadadebaze vityviT, rom isini erTmaneTis kontrarulia,

Tu ki SeuZlebelia, rom orive erTdrouliad iyos WeSmariti, magram SesaZloa

orive erTdroulad iyos mcdari.

iii. or martiv kategoriul winadadebaze vityviT, rom isini erTmaneTis

subkontrarulia, Tu ki SeuZlebelia, rom orive erTdrouliad iyos mcdari, magram

SesaZloa orive erTdroulad iyos WeSmariti.

iv. or martiv kategoriul winadadebaze vityviT, rom erTi meoris daqvemdebarebulia

rodesac pirvelis WeSmariteba aucileblad gulisxmobs meoris WeSmaritebas da

ara piruku, anu daqvemdebarebuli winadadeba SesaZloa WeSmariti iyos, xolo misi

damaqvemdebarebeli ara. Aqedan, SegviZlia vTqvaT, rom

kontrarulia SaP da SeP;kontradiqtorulia wyvilebi (SaP da SoP) da (SeP da SiP);

subkontrarulia (SiP da SoP);xolo daqvemdebarebul winadadebaTa wyvili iqneba (SaP da SiP)

da (SeP da SoP). vTqvaT, (S a P) aris A, ( S e P) aris E, ( S i P) aris I da ( S o P) aris O. axla, Tu

davxazavT kvadrats, romlos zeda kuTxeebi iqneba A da E, xolo qveda kuTxeebi,

Sesabamisad, I da O, miviRebT imas, rasac tradiciulad logikur kvadrats uwodeben da

mis SemoRebas aristoteles miaweren.

o daxazeT logikuri kvadrati da miuTiTeT msjelobebs Soris

kontraruli, kotradiqtoruli, subkontraruli da daqvemdebarebis

mimarTebebi. ra mimarTeba gveqneba diagonalze?

o ageT martivi kategoriuli winadadebebi Semdegi teminebidan:ricxvi,

dadebiTi, simravle, qvesimravle, biologiuri arseba, mokvdavi,

ukvdavi. gansazRvreT venis diagramebis meTodiT, romeli iqneba

WeSmariti.

62

o aageT WeSmaritebis cxrili martivi kategoriuli winadadebebisaTvis. venis diagramebis meTodis gamoyenebiT daasabuTeT am cxrilis marTebuloba.

o daasabuTeT imave cxrilis marTebuloba simravlisTeoriuli kanonebis gamoyenebiT.

o imave meTodebiT, rac me-(4.3) da me-4.4(3) svarjiSoebSia miTiTebuli gansazRvreT (i) – (iv) pirobebis samarTlianoba.

o logikur kvadretSi miTiTebuli mimarTebebi daamtkiceT rogorc predikatTa logikis Teoremebi.

damateba 3. ra wavikiTxoT?

savaldebulo wasakiTxi

l.mWedliSvili, n.ivaniZe. logika. Tb.2003 sagnis Rrma SeswavlisaTvis

1. a.tarski. logikisa da deduqciur mecnierebaTa Sesavali. Tb. 1976 2. s. klini. metamaTematikis Sesavali. niu iorki, 1952; moskovi, 1957 3. s. klini. maTematikuri logika. niu iorki, 1967; moskovi, 1973 4. a. CerCi. maTematikuri logikis Sesavali. prinstoni, 1956; moskovi, 1960 5. e. mendelsoni. maTematikuri logikis Sesavali. niu iorki, 1963;

moskovi, 1976 6. h. kari. maTematikuri logikis safuZvlebi. niu iorki, 1963; moskovi, 1969 7. v. quaini. logikis meTodebi. niu iorki, 1955 8. v. quaini. logikis filosofia. niu iorki, 1970; moskovi, 2008 9. a. frenkeli, i.bar-hileli. simravleTa Teoriis safuZvlebi.

amsterdami, 1958; moskovi, 1966

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